{"id":714,"date":"2025-07-14T21:26:18","date_gmt":"2025-07-14T21:26:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=714"},"modified":"2026-01-16T21:17:55","modified_gmt":"2026-01-16T21:17:55","slug":"module-18-parametric-functions-and-vectors-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/module-18-parametric-functions-and-vectors-cheat-sheet\/","title":{"raw":"Parametric Functions and Vectors: Cheat Sheet","rendered":"Parametric Functions and Vectors: Cheat Sheet"},"content":{"raw":"<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Essential Concepts<\/h2>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Parametric Equations<\/h3>\r\n<ul>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Parametric equations are a set of equations where both [latex]x[\/latex] and [latex]y[\/latex] are expressed as functions of a third variable [latex]t[\/latex] (the parameter, often representing time): [latex]x = f(t)[\/latex] and [latex]y = g(t)[\/latex]. The set of ordered pairs [latex](x(t), y(t))[\/latex] forms a plane curve.<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Creating Parametric Equations from Rectangular Form<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Given a rectangular equation like [latex]y = x^2 - 1[\/latex]:<\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Choose a parameterization for [latex]x[\/latex], typically [latex]x(t) = t[\/latex] (simplest choice)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Substitute this into the rectangular equation to get [latex]y(t)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Example: If [latex]x(t) = t[\/latex], then [latex]y(t) = t^2 - 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">There are infinitely many ways to parameterize the same curve<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Graphing Parametric Equations<\/strong>:<\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Create a table with columns for [latex]t[\/latex], [latex]x(t)[\/latex], and [latex]y(t)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Choose values for [latex]t[\/latex] in increasing order<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Calculate corresponding [latex]x[\/latex] and [latex]y[\/latex] values<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Plot the [latex](x, y)[\/latex] points<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Draw arrows showing the direction as [latex]t[\/latex] increases (orientation)<\/li>\r\n<\/ol>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Converting parametric equations to rectangular form:<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><em>For Polynomial, Exponential, or Logarithmic Equations<\/em>:<\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Solve the simpler equation for [latex]t[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Substitute this expression for [latex]t[\/latex] into the other equation<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Simplify to get an equation in [latex]x[\/latex] and [latex]y[\/latex] only<\/li>\r\n<\/ol>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><em>For Trigonometric Equations<\/em>: When [latex]x(t) = a\\cos t[\/latex] and [latex]y(t) = b\\sin t[\/latex]:<\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Solve for [latex]\\cos t[\/latex] and [latex]\\sin t[\/latex]: [latex]\\frac{x}{a} = \\cos t[\/latex] and [latex]\\frac{y}{b} = \\sin t[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Use the Pythagorean identity: [latex]\\cos^2 t + \\sin^2 t = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Substitute: [latex]\\left(\\frac{x}{a}\\right)^2 + \\left(\\frac{y}{b}\\right)^2 = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">This produces [latex]\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1[\/latex] (an ellipse)<\/li>\r\n<\/ol>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">When eliminating the parameter, be careful to preserve any domain restrictions from the original parametric equations.<\/p>\r\n\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Graphing Parametric Equations<\/h3>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Key Features to Identify<\/strong>:<\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\"><strong>Orientation<\/strong>: The direction along the curve as [latex]t[\/latex] increases, shown with arrows<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\"><strong>Starting and ending points<\/strong>: Evaluate at the minimum and maximum values of [latex]t[\/latex] in the given interval<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\"><strong>Special points<\/strong>: Where the curve crosses axes or has interesting behavior<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Comparing Parametric and Rectangular Graphs<\/strong>: The same curve can be represented both ways. Parametric form shows direction\/motion; rectangular form shows the geometric shape. When graphed together, they produce identical curves.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Projectile Motion<\/strong> - A key application of parametric equations:<\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Horizontal position: [latex]x = (v_0 \\cos\\theta)t[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Vertical position: [latex]y = -16t^2 + (v_0 \\sin\\theta)t + h[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Where [latex]v_0[\/latex] is initial velocity, [latex]\\theta[\/latex] is launch angle, and [latex]h[\/latex] is initial height. The [latex]-16[\/latex] comes from gravity (in feet per second squared).<\/p>\r\n\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Understanding Vectors<\/h3>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A vector is a quantity with both magnitude (size\/length) and direction, represented as a directed line segment with an arrow.<\/p>\r\n\r\n<ul>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Vector Components:\r\n<ul>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Initial point: Where the vector starts<\/li>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Terminal point: Where the vector ends (arrow tip)<\/li>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Position vector: A vector with initial point at the origin [latex](0, 0)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Vector Notation<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">:<\/span>\r\n<ul>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Boldface: [latex]\\mathbf{v}[\/latex] or with arrow: [latex]\\vec{v}[\/latex]<\/li>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">From point [latex]P[\/latex] to point [latex]Q[\/latex]: [latex]\\overrightarrow{PQ}[\/latex]<\/li>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Component form: [latex]\\langle a, b \\rangle[\/latex] or [latex]v = a\\mathbf{i} + b\\mathbf{j}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Position Vector: Given initial point [latex](x_1, y_1)[\/latex] and terminal point [latex](x_2, y_2)[\/latex], the position vector is: [latex]\\mathbf{v} = \\langle x_2 - x_1, y_2 - y_1 \\rangle[\/latex]<\/li>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Magnitude (Length): For vector [latex]\\mathbf{v} = \\langle a, b \\rangle[\/latex]: [latex]|\\mathbf{v}| = \\sqrt{a^2 + b^2}[\/latex]\r\n<ul>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Use the Pythagorean theorem\u2014the magnitude is the distance from initial to terminal point.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Direction (Angle): The angle [latex]\\theta[\/latex] from the positive x-axis: [latex]\\tan\\theta = \\frac{b}{a}[\/latex], so [latex]\\theta = \\tan^{-1}\\left(\\frac{b}{a}\\right)[\/latex]\r\n<ul>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Check which quadrant the vector is in, as inverse tangent only gives angles in quadrants I and IV.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Component Form<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">: Every vector can be written as [latex]\\mathbf{v} = a\\mathbf{i} + b\\mathbf{j}[\/latex] where <\/span>[latex]\\mathbf{i} = \\langle 1, 0 \\rangle[\/latex] is the horizontal unit vector, [latex]\\mathbf{j} = \\langle 0, 1 \\rangle[\/latex] is the vertical unit vector, [latex]a[\/latex] is the horizontal component, and [latex]b[\/latex] is the vertical component<\/li>\r\n \t<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Unit Vector: A vector with magnitude 1. To find a unit vector in the direction of [latex]\\mathbf{v}[\/latex]: [latex]\\frac{\\mathbf{v}}{|\\mathbf{v}|}[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Writing Vectors in Terms of Magnitude and Direction<\/strong>: [latex]\\mathbf{v} = |\\mathbf{v}|\\cos\\theta\\mathbf{i} + |\\mathbf{v}|\\sin\\theta\\mathbf{j}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Where [latex]|\\mathbf{v}|[\/latex] is the magnitude and [latex]\\theta[\/latex] is the direction angle. This connects to [latex]x = |\\mathbf{v}|\\cos\\theta[\/latex] and [latex]y = |\\mathbf{v}|\\sin\\theta[\/latex].<\/p>\r\n\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Operations with Vectors<\/h3>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Vector Addition<\/strong> - To find [latex]\\mathbf{u} + \\mathbf{v}[\/latex]:<\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\"><strong>Geometrically<\/strong>: Place the initial point of [latex]\\mathbf{v}[\/latex] at the terminal point of [latex]\\mathbf{u}[\/latex]. The sum goes from the start of [latex]\\mathbf{u}[\/latex] to the end of [latex]\\mathbf{v}[\/latex].<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\"><strong>Algebraically<\/strong>: Add corresponding components [latex]\\langle a, b \\rangle + \\langle c, d \\rangle = \\langle a+c, b+d \\rangle[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">In [latex]\\mathbf{i}, \\mathbf{j}[\/latex] notation: [latex](a\\mathbf{i} + b\\mathbf{j}) + (c\\mathbf{i} + d\\mathbf{j}) = (a+c)\\mathbf{i} + (b+d)\\mathbf{j}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Vector Subtraction<\/strong> - To find [latex]\\mathbf{u} - \\mathbf{v}[\/latex]:<\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">View as [latex]\\mathbf{u} + (-\\mathbf{v})[\/latex] where [latex]-\\mathbf{v}[\/latex] reverses the direction of [latex]\\mathbf{v}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\"><strong>Algebraically<\/strong>: Subtract corresponding components [latex]\\langle a, b \\rangle - \\langle c, d \\rangle = \\langle a-c, b-d \\rangle[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Scalar Multiplication<\/strong> - Multiplying a vector by a constant [latex]k[\/latex]:<\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Changes the magnitude by factor [latex]|k|[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">If [latex]k &lt; 0[\/latex], reverses direction<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Does not change direction if [latex]k &gt; 0[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[latex]k\\langle a, b \\rangle = \\langle ka, kb \\rangle[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">In [latex]\\mathbf{i}, \\mathbf{j}[\/latex] notation: [latex]k(a\\mathbf{i} + b\\mathbf{j}) = ka\\mathbf{i} + kb\\mathbf{j}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Properties<\/strong>: Scalar multiplication and vector addition can be combined: [latex]k\\mathbf{u} + m\\mathbf{v} = k\\langle a, b \\rangle + m\\langle c, d \\rangle = \\langle ka + mc, kb + md \\rangle[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Dot Product<\/strong> - Multiplying two vectors to get a scalar: [latex]\\mathbf{v} \\cdot \\mathbf{u} = \\langle a, b \\rangle \\cdot \\langle c, d \\rangle = ac + bd[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Multiply corresponding components and add the results. The result is a <strong>number<\/strong>, not a vector.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Finding the Angle Between Two Vectors<\/strong> using dot product: [latex]\\cos\\theta = \\frac{\\mathbf{v}}{|\\mathbf{v}|} \\cdot \\frac{\\mathbf{u}}{|\\mathbf{u}|}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">This gives: [latex]\\theta = \\cos^{-1}\\left(\\frac{\\mathbf{v} \\cdot \\mathbf{u}}{|\\mathbf{v}||\\mathbf{u}|}\\right)[\/latex]<\/p>\r\n\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Key Equations<\/h2>\r\n<table style=\"width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Position vector<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]\\mathbf{v} = \\langle x_2 - x_1, y_2 - y_1 \\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Vector magnitude<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]|\\mathbf{v}| = \\sqrt{a^2 + b^2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Direction angle<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]\\tan\\theta = \\frac{b}{a}[\/latex], so [latex]\\theta = \\tan^{-1}\\left(\\frac{b}{a}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Unit vector<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]\\frac{\\mathbf{v}}{|\\mathbf{v}|}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Vector in terms of magnitude and direction<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]\\mathbf{v} = |\\mathbf{v}|\\cos\\theta\\mathbf{i} + |\\mathbf{v}|\\sin\\theta\\mathbf{j}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Components from magnitude and direction<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]x = |\\mathbf{v}|\\cos\\theta[\/latex] and [latex]y = |\\mathbf{v}|\\sin\\theta[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Vector addition<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]\\mathbf{v} + \\mathbf{u} = \\langle a+c, b+d \\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Vector subtraction<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]\\mathbf{v} - \\mathbf{u} = \\langle a-c, b-d \\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Scalar multiplication<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]k\\mathbf{v} = \\langle ka, kb \\rangle[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Dot product<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]\\mathbf{v} \\cdot \\mathbf{u} = ac + bd[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Angle between vectors<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]\\cos\\theta = \\frac{\\mathbf{v} \\cdot \\mathbf{u}}{|\\mathbf{v}||\\mathbf{u}|}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Projectile motion (horizontal)<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]x = (v_0\\cos\\theta)t[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.662%;\"><strong>Projectile motion (vertical)<\/strong><\/td>\r\n<td style=\"width: 54.9296%;\">[latex]y = -16t^2 + (v_0\\sin\\theta)t + h[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Glossary<\/h2>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>component form<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A way of writing a vector showing its horizontal and vertical parts: [latex]\\langle a, b \\rangle[\/latex] or [latex]a\\mathbf{i} + b\\mathbf{j}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>curvilinear path<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A curved path or trajectory, as opposed to a straight line; the type of motion described by parametric equations.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>dot product<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The sum of the products of corresponding components of two vectors; produces a scalar: [latex]\\mathbf{v} \\cdot \\mathbf{u} = ac + bd[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>initial point<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The starting point of a vector; the origin [latex](0, 0)[\/latex] for a position vector.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>magnitude<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The length of a vector, calculated using the Pythagorean theorem: [latex]|\\mathbf{v}| = \\sqrt{a^2 + b^2}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>orientation<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The direction along a parametric curve as the parameter [latex]t[\/latex] increases, typically shown with arrows.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>parameter<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A third variable (often [latex]t[\/latex] for time) upon which both [latex]x[\/latex] and [latex]y[\/latex] depend in parametric equations.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>parametric equations<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A pair of equations [latex]x = f(t)[\/latex] and [latex]y = g(t)[\/latex] that define a curve by expressing both coordinates as functions of a parameter.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>position vector<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A vector with its initial point at the origin [latex](0, 0)[\/latex] and terminal point at [latex]\\langle a, b \\rangle[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>resultant<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The vector that results from adding or subtracting two vectors, or from scalar multiplication.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>scalar<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A quantity with magnitude but no direction; a constant or real number.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>scalar multiplication<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The operation of multiplying a vector by a scalar, which changes the magnitude and possibly reverses direction: [latex]k\\mathbf{v} = \\langle ka, kb \\rangle[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>standard position<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The placement of a vector with initial point at [latex](0, 0)[\/latex]; the position vector representation.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>terminal point<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The ending point of a vector, indicated by the arrowhead.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>unit vector<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A vector with magnitude 1; [latex]\\mathbf{i} = \\langle 1, 0 \\rangle[\/latex] and [latex]\\mathbf{j} = \\langle 0, 1 \\rangle[\/latex] are the standard unit vectors.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>vector<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A quantity with both magnitude and direction, represented as a directed line segment.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>vector addition<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The operation of combining two vectors by adding corresponding components to produce a resultant vector.<\/p>","rendered":"<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Essential Concepts<\/h2>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Parametric Equations<\/h3>\n<ul>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Parametric equations are a set of equations where both [latex]x[\/latex] and [latex]y[\/latex] are expressed as functions of a third variable [latex]t[\/latex] (the parameter, often representing time): [latex]x = f(t)[\/latex] and [latex]y = g(t)[\/latex]. The set of ordered pairs [latex](x(t), y(t))[\/latex] forms a plane curve.<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Creating Parametric Equations from Rectangular Form<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Given a rectangular equation like [latex]y = x^2 - 1[\/latex]:<\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Choose a parameterization for [latex]x[\/latex], typically [latex]x(t) = t[\/latex] (simplest choice)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Substitute this into the rectangular equation to get [latex]y(t)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Example: If [latex]x(t) = t[\/latex], then [latex]y(t) = t^2 - 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">There are infinitely many ways to parameterize the same curve<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Graphing Parametric Equations<\/strong>:<\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Create a table with columns for [latex]t[\/latex], [latex]x(t)[\/latex], and [latex]y(t)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Choose values for [latex]t[\/latex] in increasing order<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Calculate corresponding [latex]x[\/latex] and [latex]y[\/latex] values<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Plot the [latex](x, y)[\/latex] points<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Draw arrows showing the direction as [latex]t[\/latex] increases (orientation)<\/li>\n<\/ol>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Converting parametric equations to rectangular form:<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><em>For Polynomial, Exponential, or Logarithmic Equations<\/em>:<\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Solve the simpler equation for [latex]t[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Substitute this expression for [latex]t[\/latex] into the other equation<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Simplify to get an equation in [latex]x[\/latex] and [latex]y[\/latex] only<\/li>\n<\/ol>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><em>For Trigonometric Equations<\/em>: When [latex]x(t) = a\\cos t[\/latex] and [latex]y(t) = b\\sin t[\/latex]:<\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Solve for [latex]\\cos t[\/latex] and [latex]\\sin t[\/latex]: [latex]\\frac{x}{a} = \\cos t[\/latex] and [latex]\\frac{y}{b} = \\sin t[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Use the Pythagorean identity: [latex]\\cos^2 t + \\sin^2 t = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Substitute: [latex]\\left(\\frac{x}{a}\\right)^2 + \\left(\\frac{y}{b}\\right)^2 = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">This produces [latex]\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1[\/latex] (an ellipse)<\/li>\n<\/ol>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">When eliminating the parameter, be careful to preserve any domain restrictions from the original parametric equations.<\/p>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Graphing Parametric Equations<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Key Features to Identify<\/strong>:<\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\"><strong>Orientation<\/strong>: The direction along the curve as [latex]t[\/latex] increases, shown with arrows<\/li>\n<li class=\"whitespace-normal break-words pl-2\"><strong>Starting and ending points<\/strong>: Evaluate at the minimum and maximum values of [latex]t[\/latex] in the given interval<\/li>\n<li class=\"whitespace-normal break-words pl-2\"><strong>Special points<\/strong>: Where the curve crosses axes or has interesting behavior<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Comparing Parametric and Rectangular Graphs<\/strong>: The same curve can be represented both ways. Parametric form shows direction\/motion; rectangular form shows the geometric shape. When graphed together, they produce identical curves.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Projectile Motion<\/strong> &#8211; A key application of parametric equations:<\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Horizontal position: [latex]x = (v_0 \\cos\\theta)t[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Vertical position: [latex]y = -16t^2 + (v_0 \\sin\\theta)t + h[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Where [latex]v_0[\/latex] is initial velocity, [latex]\\theta[\/latex] is launch angle, and [latex]h[\/latex] is initial height. The [latex]-16[\/latex] comes from gravity (in feet per second squared).<\/p>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Understanding Vectors<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A vector is a quantity with both magnitude (size\/length) and direction, represented as a directed line segment with an arrow.<\/p>\n<ul>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Vector Components:\n<ul>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Initial point: Where the vector starts<\/li>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Terminal point: Where the vector ends (arrow tip)<\/li>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Position vector: A vector with initial point at the origin [latex](0, 0)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Vector Notation<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">:<\/span>\n<ul>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Boldface: [latex]\\mathbf{v}[\/latex] or with arrow: [latex]\\vec{v}[\/latex]<\/li>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">From point [latex]P[\/latex] to point [latex]Q[\/latex]: [latex]\\overrightarrow{PQ}[\/latex]<\/li>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Component form: [latex]\\langle a, b \\rangle[\/latex] or [latex]v = a\\mathbf{i} + b\\mathbf{j}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Position Vector: Given initial point [latex](x_1, y_1)[\/latex] and terminal point [latex](x_2, y_2)[\/latex], the position vector is: [latex]\\mathbf{v} = \\langle x_2 - x_1, y_2 - y_1 \\rangle[\/latex]<\/li>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Magnitude (Length): For vector [latex]\\mathbf{v} = \\langle a, b \\rangle[\/latex]: [latex]|\\mathbf{v}| = \\sqrt{a^2 + b^2}[\/latex]\n<ul>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Use the Pythagorean theorem\u2014the magnitude is the distance from initial to terminal point.<\/li>\n<\/ul>\n<\/li>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Direction (Angle): The angle [latex]\\theta[\/latex] from the positive x-axis: [latex]\\tan\\theta = \\frac{b}{a}[\/latex], so [latex]\\theta = \\tan^{-1}\\left(\\frac{b}{a}\\right)[\/latex]\n<ul>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Check which quadrant the vector is in, as inverse tangent only gives angles in quadrants I and IV.<\/li>\n<\/ul>\n<\/li>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Component Form<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">: Every vector can be written as [latex]\\mathbf{v} = a\\mathbf{i} + b\\mathbf{j}[\/latex] where <\/span>[latex]\\mathbf{i} = \\langle 1, 0 \\rangle[\/latex] is the horizontal unit vector, [latex]\\mathbf{j} = \\langle 0, 1 \\rangle[\/latex] is the vertical unit vector, [latex]a[\/latex] is the horizontal component, and [latex]b[\/latex] is the vertical component<\/li>\n<li class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Unit Vector: A vector with magnitude 1. To find a unit vector in the direction of [latex]\\mathbf{v}[\/latex]: [latex]\\frac{\\mathbf{v}}{|\\mathbf{v}|}[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Writing Vectors in Terms of Magnitude and Direction<\/strong>: [latex]\\mathbf{v} = |\\mathbf{v}|\\cos\\theta\\mathbf{i} + |\\mathbf{v}|\\sin\\theta\\mathbf{j}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Where [latex]|\\mathbf{v}|[\/latex] is the magnitude and [latex]\\theta[\/latex] is the direction angle. This connects to [latex]x = |\\mathbf{v}|\\cos\\theta[\/latex] and [latex]y = |\\mathbf{v}|\\sin\\theta[\/latex].<\/p>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Operations with Vectors<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Vector Addition<\/strong> &#8211; To find [latex]\\mathbf{u} + \\mathbf{v}[\/latex]:<\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\"><strong>Geometrically<\/strong>: Place the initial point of [latex]\\mathbf{v}[\/latex] at the terminal point of [latex]\\mathbf{u}[\/latex]. The sum goes from the start of [latex]\\mathbf{u}[\/latex] to the end of [latex]\\mathbf{v}[\/latex].<\/li>\n<li class=\"whitespace-normal break-words pl-2\"><strong>Algebraically<\/strong>: Add corresponding components [latex]\\langle a, b \\rangle + \\langle c, d \\rangle = \\langle a+c, b+d \\rangle[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">In [latex]\\mathbf{i}, \\mathbf{j}[\/latex] notation: [latex](a\\mathbf{i} + b\\mathbf{j}) + (c\\mathbf{i} + d\\mathbf{j}) = (a+c)\\mathbf{i} + (b+d)\\mathbf{j}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Vector Subtraction<\/strong> &#8211; To find [latex]\\mathbf{u} - \\mathbf{v}[\/latex]:<\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">View as [latex]\\mathbf{u} + (-\\mathbf{v})[\/latex] where [latex]-\\mathbf{v}[\/latex] reverses the direction of [latex]\\mathbf{v}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\"><strong>Algebraically<\/strong>: Subtract corresponding components [latex]\\langle a, b \\rangle - \\langle c, d \\rangle = \\langle a-c, b-d \\rangle[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Scalar Multiplication<\/strong> &#8211; Multiplying a vector by a constant [latex]k[\/latex]:<\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Changes the magnitude by factor [latex]|k|[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">If [latex]k < 0[\/latex], reverses direction<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Does not change direction if [latex]k > 0[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">[latex]k\\langle a, b \\rangle = \\langle ka, kb \\rangle[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">In [latex]\\mathbf{i}, \\mathbf{j}[\/latex] notation: [latex]k(a\\mathbf{i} + b\\mathbf{j}) = ka\\mathbf{i} + kb\\mathbf{j}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Properties<\/strong>: Scalar multiplication and vector addition can be combined: [latex]k\\mathbf{u} + m\\mathbf{v} = k\\langle a, b \\rangle + m\\langle c, d \\rangle = \\langle ka + mc, kb + md \\rangle[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Dot Product<\/strong> &#8211; Multiplying two vectors to get a scalar: [latex]\\mathbf{v} \\cdot \\mathbf{u} = \\langle a, b \\rangle \\cdot \\langle c, d \\rangle = ac + bd[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Multiply corresponding components and add the results. The result is a <strong>number<\/strong>, not a vector.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Finding the Angle Between Two Vectors<\/strong> using dot product: [latex]\\cos\\theta = \\frac{\\mathbf{v}}{|\\mathbf{v}|} \\cdot \\frac{\\mathbf{u}}{|\\mathbf{u}|}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">This gives: [latex]\\theta = \\cos^{-1}\\left(\\frac{\\mathbf{v} \\cdot \\mathbf{u}}{|\\mathbf{v}||\\mathbf{u}|}\\right)[\/latex]<\/p>\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Key Equations<\/h2>\n<table style=\"width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Position vector<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]\\mathbf{v} = \\langle x_2 - x_1, y_2 - y_1 \\rangle[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Vector magnitude<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]|\\mathbf{v}| = \\sqrt{a^2 + b^2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Direction angle<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]\\tan\\theta = \\frac{b}{a}[\/latex], so [latex]\\theta = \\tan^{-1}\\left(\\frac{b}{a}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Unit vector<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]\\frac{\\mathbf{v}}{|\\mathbf{v}|}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Vector in terms of magnitude and direction<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]\\mathbf{v} = |\\mathbf{v}|\\cos\\theta\\mathbf{i} + |\\mathbf{v}|\\sin\\theta\\mathbf{j}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Components from magnitude and direction<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]x = |\\mathbf{v}|\\cos\\theta[\/latex] and [latex]y = |\\mathbf{v}|\\sin\\theta[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Vector addition<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]\\mathbf{v} + \\mathbf{u} = \\langle a+c, b+d \\rangle[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Vector subtraction<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]\\mathbf{v} - \\mathbf{u} = \\langle a-c, b-d \\rangle[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Scalar multiplication<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]k\\mathbf{v} = \\langle ka, kb \\rangle[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Dot product<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]\\mathbf{v} \\cdot \\mathbf{u} = ac + bd[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Angle between vectors<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]\\cos\\theta = \\frac{\\mathbf{v} \\cdot \\mathbf{u}}{|\\mathbf{v}||\\mathbf{u}|}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Projectile motion (horizontal)<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]x = (v_0\\cos\\theta)t[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.662%;\"><strong>Projectile motion (vertical)<\/strong><\/td>\n<td style=\"width: 54.9296%;\">[latex]y = -16t^2 + (v_0\\sin\\theta)t + h[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Glossary<\/h2>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>component form<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A way of writing a vector showing its horizontal and vertical parts: [latex]\\langle a, b \\rangle[\/latex] or [latex]a\\mathbf{i} + b\\mathbf{j}[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>curvilinear path<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A curved path or trajectory, as opposed to a straight line; the type of motion described by parametric equations.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>dot product<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The sum of the products of corresponding components of two vectors; produces a scalar: [latex]\\mathbf{v} \\cdot \\mathbf{u} = ac + bd[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>initial point<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The starting point of a vector; the origin [latex](0, 0)[\/latex] for a position vector.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>magnitude<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The length of a vector, calculated using the Pythagorean theorem: [latex]|\\mathbf{v}| = \\sqrt{a^2 + b^2}[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>orientation<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The direction along a parametric curve as the parameter [latex]t[\/latex] increases, typically shown with arrows.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>parameter<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A third variable (often [latex]t[\/latex] for time) upon which both [latex]x[\/latex] and [latex]y[\/latex] depend in parametric equations.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>parametric equations<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A pair of equations [latex]x = f(t)[\/latex] and [latex]y = g(t)[\/latex] that define a curve by expressing both coordinates as functions of a parameter.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>position vector<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A vector with its initial point at the origin [latex](0, 0)[\/latex] and terminal point at [latex]\\langle a, b \\rangle[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>resultant<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The vector that results from adding or subtracting two vectors, or from scalar multiplication.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>scalar<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A quantity with magnitude but no direction; a constant or real number.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>scalar multiplication<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The operation of multiplying a vector by a scalar, which changes the magnitude and possibly reverses direction: [latex]k\\mathbf{v} = \\langle ka, kb \\rangle[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>standard position<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The placement of a vector with initial point at [latex](0, 0)[\/latex]; the position vector representation.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>terminal point<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The ending point of a vector, indicated by the arrowhead.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>unit vector<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A vector with magnitude 1; [latex]\\mathbf{i} = \\langle 1, 0 \\rangle[\/latex] and [latex]\\mathbf{j} = \\langle 0, 1 \\rangle[\/latex] are the standard unit vectors.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>vector<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A quantity with both magnitude and direction, represented as a directed line segment.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>vector addition<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The operation of combining two vectors by adding corresponding components to produce a resultant vector.<\/p>\n","protected":false},"author":67,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":520,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/714"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/714\/revisions"}],"predecessor-version":[{"id":5419,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/714\/revisions\/5419"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/520"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/714\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=714"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=714"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=714"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=714"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}