{"id":486,"date":"2025-10-09T15:13:54","date_gmt":"2025-10-09T15:13:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=486"},"modified":"2025-10-14T15:33:37","modified_gmt":"2025-10-14T15:33:37","slug":"foundations-of-vectors-and-3d-space-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/foundations-of-vectors-and-3d-space-cheat-sheet\/","title":{"raw":"Foundations of Vectors and 3D Space: Cheat Sheet","rendered":"Foundations of Vectors and 3D Space: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<strong>Vector Basics<\/strong>\r\n<ul id=\"fs-id1167793944779\" data-bullet-style=\"bullet\">\r\n \t<li>Vectors are used to represent quantities that have both magnitude and direction.<\/li>\r\n \t<li>We can add vectors by using the parallelogram method or the triangle method to find the sum. We can multiply a vector by a scalar to change its length or give it the opposite direction.<\/li>\r\n \t<li>Subtraction of vectors is defined in terms of adding the negative of the vector.<\/li>\r\n \t<li>A vector is written in component form as\u00a0[latex]{\\bf{v}}=\\langle{x,y}\\rangle[\/latex].<\/li>\r\n \t<li>The magnitude of a vector is a scalar:\u00a0[latex]\\parallel{\\bf{v}}\\parallel=\\sqrt{x^2+y^2}[\/latex].<\/li>\r\n \t<li>A unit vector [latex]\\bf{u}[\/latex] has magnitude [latex]1[\/latex] and can be found by dividing a vector by its magnitude:\u00a0[latex]{\\bf{u}}=\\frac{1}{\\parallel{\\bf{v}}\\parallel}{\\bf{v}}[\/latex] \u00a0The standard unit vectors are\u00a0[latex]{\\bf{i}}=\\langle{1,0}\\rangle[\/latex] and\u00a0[latex]{\\bf{j}}=\\langle{0,1}\\rangle[\/latex]. A vector [latex]{\\bf{v}}=\\langle{x,y}\\rangle[\/latex] can be expressed in terms of the standard unit vectors as\u00a0[latex]{\\bf{v}}=x{\\bf{i}}+y{\\bf{j}}[\/latex].<\/li>\r\n<\/ul>\r\n<div class=\"os-section-area\"><section id=\"fs-id1167794326123\" class=\"key-concepts\" data-depth=\"1\">\r\n<ul id=\"fs-id1167793944779\" data-bullet-style=\"bullet\">\r\n \t<li>Vectors are often used in physics and engineering to represent forces and velocities, among other quantities.<\/li>\r\n<\/ul>\r\n<strong>Three-Dimensional Space<\/strong>\r\n<ul>\r\n \t<li>The three-dimensional coordinate system is built around a set of three axes that intersect at right angles at a single point, the origin.\u00a0Ordered triples [latex](x,y,z)[\/latex]\u00a0are used to describe the location of a point in space.<\/li>\r\n \t<li>The distance [latex]d[\/latex]\u00a0between points [latex](x_{1},y_{1},z_{1})[\/latex] and\u00a0[latex](x_{2},y_{2},z_{2})[\/latex]\u00a0is given by the formula [latex]d=\\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}[\/latex].<\/li>\r\n \t<li>In three dimensions, the equations [latex]x=a[\/latex],\u00a0[latex]y=b[\/latex], and\u00a0[latex]z=c[\/latex]\u00a0describe planes that are parallel to the coordinate planes.<\/li>\r\n \t<li>The standard equation of a sphere with center [latex](a,b,c)[\/latex] and radius [latex]r[\/latex]\u00a0is [latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex].<\/li>\r\n \t<li>In three dimensions, as in two, vectors are commonly expressed in component form, [latex]{\\bf{v}}=\\langle{x,y,z}\\rangle[\/latex], or in terms of the standard unit vectors,\u00a0[latex]x{\\bf{i}}+y{\\bf{j}}+z{\\bf{k}}[\/latex].<\/li>\r\n \t<li>Properties of vectors in space are a natural extension of the properties for vectors in a plane. Let\u00a0[latex]{\\bf{v}}=\\langle{x_{1},y_{1},z_{1}}\\rangle[\/latex] and\u00a0[latex]{\\bf{w}}=\\langle{x_{2},y_{2},z_{2}}\\rangle[\/latex]\u00a0be vectors, and let [latex]k[\/latex]\u00a0be a scalar.\r\n<ul>\r\n \t<li>Scalar multiplication:\u00a0[latex]k{\\bf{v}}=\\langle{kx_{1},ky_{1},kz_{1}}\\rangle[\/latex]<\/li>\r\n \t<li>Vector addition:\u00a0[latex]{\\bf{v}}+{\\bf{w}}=\\langle{x_{1},y_{1},z_{1}}\\rangle+\\langle{x_{2},y_{2},z_{2}}\\rangle=\\langle{x_{1}+x_{2},y_{1}+y_{2},z_{1}+z_{2}}\\rangle[\/latex]<\/li>\r\n \t<li>Vector subtraction:\u00a0[latex]{\\bf{v}}-{\\bf{w}}=\\langle{x_{1},y_{1},z_{1}}\\rangle-\\langle{x_{2},y_{2},z_{2}}\\rangle=\\langle{x_{1}-x_{2},y_{1}-y_{2},z_{1}-z_{2}}\\rangle[\/latex]<\/li>\r\n \t<li>Vector magnitude: [latex]\\parallel{\\bf{v}}\\parallel=\\sqrt{{x_{1}}^{2}+{y_{1}}^{2}+{z_{1}}^{2}}[\/latex]<\/li>\r\n \t<li>Unit vector in the direction of [latex]{\\bf{v}}[\/latex]: [latex]\\frac{\\bf{v}}{\\parallel{\\bf{v}}\\parallel}=\\frac{1}{\\parallel{\\bf{v}}\\parallel}\\langle{x_{1},y_{1},z_{1}}\\rangle=\\langle{\\frac{x_{1}}{\\parallel{\\bf{v}}\\parallel},\\frac{y_{1}}{\\parallel{\\bf{v}}\\parallel},\\frac{z_{1}}{\\parallel{\\bf{v}}\\parallel}}\\rangle,{\\bf{v}}\\ne{0}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/section><strong>Vector Operations in Space<\/strong>\r\n\r\n<\/div>\r\n<ul>\r\n \t<li>The dot product, or scalar product, of two vectors\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex] is\u00a0[latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex].<\/li>\r\n \t<li>The dot product satisfies the following properties:\r\n<ul>\r\n \t<li>[latex]{\\bf{u}}\\cdot{\\bf{v}}={\\bf{v}}\\cdot{\\bf{u}}[\/latex]<\/li>\r\n \t<li>[latex]{\\bf{u}}\\cdot\\left({\\bf{v}}+{\\bf{w}}\\right)={\\bf{u}}\\cdot{\\bf{v}}+{\\bf{u}}\\cdot{\\bf{w}}[\/latex]<\/li>\r\n \t<li>[latex]c({\\bf{u}}\\cdot{\\bf{v}})=(c{\\bf{u}})\\cdot{\\bf{v}}={\\bf{u}}\\cdot(c{\\bf{v}})[\/latex]<\/li>\r\n \t<li>[latex]{\\bf{v}}\\cdot{\\bf{v}}=\\parallel{\\bf{v}}\\parallel^{2}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The dot product of two vectors can be expressed, alternatively, as [latex]{\\bf{u}}\\cdot{\\bf{v}}=\\parallel{\\bf{u}}\\parallel\\parallel{\\bf{v}}\\parallel\\cos\\theta[\/latex].\u00a0This form of the dot product is useful for finding the measure of the angle formed by two vectors.<\/li>\r\n \t<li>Vectors\u00a0[latex]\\bf{u}[\/latex]\u00a0and\u00a0[latex]\\bf{v}[\/latex]\u00a0are orthogonal if [latex]{\\bf{u}}\\cdot{\\bf{v}}=0[\/latex].<\/li>\r\n \t<li>The angles formed by a nonzero vector and the coordinate axes are called the\u00a0<em data-effect=\"italics\">direction angles<\/em>\u00a0for the vector. The cosines of these angles are known as the\u00a0<em data-effect=\"italics\">direction cosines<\/em>.<\/li>\r\n \t<li>The vector projection of\u00a0[latex]\\bf{v}[\/latex]\u00a0onto\u00a0[latex]\\bf{u}[\/latex] is the vector [latex]\\text{proj}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel^{2}}{\\bf{u}}[\/latex].\u00a0The magnitude of this vector is known as the\u00a0<em data-effect=\"italics\">scalar projection<\/em>\u00a0of [latex]\\bf{v}[\/latex]\u00a0onto\u00a0[latex]\\bf{u}[\/latex], given by\u00a0[latex]\\text{comp}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel}[\/latex].<\/li>\r\n \t<li>\r\n<div class=\"os-section-area\"><section id=\"fs-id1163723268353\" class=\"key-concepts\" data-depth=\"1\">Work is done when a force is applied to an object, causing displacement. When the force is represented by the vector [latex]\\bf{F}[\/latex]\u00a0and the displacement is represented by the vector\u00a0[latex]\\bf{s}[\/latex], then the work done [latex]W[\/latex]\u00a0is given by the formula [latex]W={\\bf{F}}\\cdot{\\bf{s}}=\\parallel{\\bf{F}}\\parallel\\parallel{\\bf{s}}\\parallel\\cos\\theta[\/latex].<\/section><\/div><\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Distance between two points in space<\/strong>\r\n[latex]d=\\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}[\/latex]<\/li>\r\n \t<li><strong>Sphere with center [latex](a,b,c)[\/latex] and radius [latex]r[\/latex]<\/strong>\r\n[latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex]<\/li>\r\n \t<li><strong>Dot product of u and v\r\n<\/strong>[latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex]<\/li>\r\n \t<li><strong>Cosine of the angle formed by u and v<\/strong>\r\n[latex]\\cos\\theta=\\frac{{\\bf{u}}\\cdot{\\bf{v}}}{\\parallel{\\bf{u}}\\parallel\\parallel{\\bf{v}}\\parallel}[\/latex]<\/li>\r\n \t<li><strong>Vector projection of v onto u<\/strong>\r\n[latex]\\text{proj}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel^{2}}{\\bf{u}}[\/latex]<\/li>\r\n \t<li><strong>Scalar projection of v onto u<\/strong>\r\n[latex]\\text{comp}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel}[\/latex]<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li><strong>Work done by a force F to move an object through displacement vector [latex]\\overrightarrow{PQ}[\/latex]<\/strong>\r\n[latex]W={\\bf{F}}\\cdot{\\bf{s}}=\\parallel{\\bf{F}}\\parallel\\parallel{\\bf{s}}\\parallel\\cos\\theta[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>component<\/dt>\r\n \t<dd>a scalar that describes either the vertical or horizontal direction of a vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>coordinate plane<\/dt>\r\n \t<dd>a plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: the [latex]xy[\/latex]-plane, [latex]xz[\/latex]-plane, or the [latex]yz[\/latex]<span style=\"font-size: 1em;\">-plane<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>direction angles<\/dt>\r\n \t<dd>the angles formed by a nonzero vector and the coordinate axes<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>direction cosines<\/dt>\r\n \t<dd>the cosines of the angles formed by a nonzero vector and the coordinate axes<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>dot product or scalar product<\/dt>\r\n \t<dd>[latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex], where\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex]\u00a0and[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>equivalent vectors<\/dt>\r\n \t<dd>vectors that have the same magnitude and the same direction<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>initial point<\/dt>\r\n \t<dd>the starting point of a vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>magnitude<\/dt>\r\n \t<dd>the length of a vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>normalization<\/dt>\r\n \t<dd>using scalar multiplication to find a unit vector with a given direction<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>octants<\/dt>\r\n \t<dd>the eight regions of space created by the coordinate planes<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>orthogonal vectors<\/dt>\r\n \t<dd>vectors that form a right angle when placed in standard position<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>parallelogram method<\/dt>\r\n \t<dd>a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of the vectors is the diagonal of that parallelogram<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>right-hand rule<\/dt>\r\n \t<dd>a common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the [latex]z[\/latex]<span style=\"font-size: 1em;\">-axis in such a way that the fingers curl from the positive [latex]x[\/latex]-axis to the positive [latex]y[\/latex]-axis, the thumb points in the direction of the positive [latex]z[\/latex]-axis<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>scalar<\/dt>\r\n \t<dd>a real number<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>scalar multiplication<\/dt>\r\n \t<dd>a vector operation that defines the product of a scalar and a vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>scalar projection<\/dt>\r\n \t<dd>the magnitude of the vector projection of a vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>sphere<\/dt>\r\n \t<dd>the set of all points equidistant from a given point known as the <em style=\"font-size: 1em;\" data-effect=\"italics\">center<\/em><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>standard equation of a sphere<\/dt>\r\n \t<dd>[latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex] describes a sphere with center [latex](a,b,c)[\/latex]<span class=\"Apple-converted-space\"><span style=\"color: #000000;\"><span style=\"caret-color: #000000; font-size: 14px; white-space: nowrap;\">\u00a0and radius [latex]r[\/latex]<\/span><\/span><\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>standard-position Vectors<\/dt>\r\n \t<dd>a vector with initial point [latex](0,0)[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>standard unit vectors<\/dt>\r\n \t<dd>unit vectors along the coordinate axes:\u00a0[latex]{\\bf{i}}=\\langle{1,0}\\rangle[\/latex], [latex]{\\bf{j}}=\\langle{0,1}\\rangle[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>terminal point<\/dt>\r\n \t<dd>the endpoint of a vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>three-dimensional rectangular coordinate system<\/dt>\r\n \t<dd>a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple\u00a0[latex](x,y,z)[\/latex]\u00a0that plots\u00a0its<span style=\"font-size: 1em;\">\u00a0location relative to the defining axes<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>triangle inequality<\/dt>\r\n \t<dd>the length of any side of a triangle is less than the sum of the lengths of the other two sides<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>triangle method<\/dt>\r\n \t<dd>a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of the first vector; the terminal point of the sum is the terminal point of the second vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>unit vector<\/dt>\r\n \t<dd>a vector with magnitude [latex]1[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector<\/dt>\r\n \t<dd>a mathematical object that has both magnitude and direction<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector addition<\/dt>\r\n \t<dd>a vector operation that defines the sum of two vectors<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector difference<\/dt>\r\n \t<dd>the vector difference\u00a0[latex]{\\bf{v}}-{\\bf{w}}[\/latex]\u00a0is defined as\u00a0[latex]{\\bf{v}}+(-{\\bf{w}})={\\bf{v}}+(-1){\\bf{w}}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector projection<\/dt>\r\n \t<dd>the component of a vector that follows a given direction<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>vector sum<\/dt>\r\n \t<dd>the sum of two vectors, [latex]{\\bf{v}}[\/latex] and [latex]{\\bf{w}}[\/latex]can be constructed graphically by placing the initial point of\u00a0[latex]{\\bf{w}}[\/latex]\u00a0at the terminal point of [latex]{\\bf{v}}[\/latex]; then the vector sum\u00a0[latex]{\\bf{v}}+{\\bf{w}}[\/latex]\u00a0is the vector with an initial point that coincides with the initial point of [latex]{\\bf{v}}[\/latex], and with a terminal point that coincides with the terminal point of\u00a0[latex]{\\bf{w}}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>work done by a force<\/dt>\r\n \t<dd>work is generally thought of as the amount of energy it takes to move an object; if we represent an applied force by a vector [latex]\\bf{F}[\/latex] and the displacement of an object by a vector [latex]\\bf{s}[\/latex], then the work done by the force is the dot product of [latex]\\bf{F}[\/latex]\u00a0and [latex]\\bf{s}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>zero vector<\/dt>\r\n \t<dd>the vector with both initial point and terminal point\u00a0[latex](0,0)[\/latex]<\/dd>\r\n<\/dl>","rendered":"<h2>Essential Concepts<\/h2>\n<p><strong>Vector Basics<\/strong><\/p>\n<ul id=\"fs-id1167793944779\" data-bullet-style=\"bullet\">\n<li>Vectors are used to represent quantities that have both magnitude and direction.<\/li>\n<li>We can add vectors by using the parallelogram method or the triangle method to find the sum. We can multiply a vector by a scalar to change its length or give it the opposite direction.<\/li>\n<li>Subtraction of vectors is defined in terms of adding the negative of the vector.<\/li>\n<li>A vector is written in component form as\u00a0[latex]{\\bf{v}}=\\langle{x,y}\\rangle[\/latex].<\/li>\n<li>The magnitude of a vector is a scalar:\u00a0[latex]\\parallel{\\bf{v}}\\parallel=\\sqrt{x^2+y^2}[\/latex].<\/li>\n<li>A unit vector [latex]\\bf{u}[\/latex] has magnitude [latex]1[\/latex] and can be found by dividing a vector by its magnitude:\u00a0[latex]{\\bf{u}}=\\frac{1}{\\parallel{\\bf{v}}\\parallel}{\\bf{v}}[\/latex] \u00a0The standard unit vectors are\u00a0[latex]{\\bf{i}}=\\langle{1,0}\\rangle[\/latex] and\u00a0[latex]{\\bf{j}}=\\langle{0,1}\\rangle[\/latex]. A vector [latex]{\\bf{v}}=\\langle{x,y}\\rangle[\/latex] can be expressed in terms of the standard unit vectors as\u00a0[latex]{\\bf{v}}=x{\\bf{i}}+y{\\bf{j}}[\/latex].<\/li>\n<\/ul>\n<div class=\"os-section-area\">\n<section id=\"fs-id1167794326123\" class=\"key-concepts\" data-depth=\"1\">\n<ul data-bullet-style=\"bullet\">\n<li>Vectors are often used in physics and engineering to represent forces and velocities, among other quantities.<\/li>\n<\/ul>\n<p><strong>Three-Dimensional Space<\/strong><\/p>\n<ul>\n<li>The three-dimensional coordinate system is built around a set of three axes that intersect at right angles at a single point, the origin.\u00a0Ordered triples [latex](x,y,z)[\/latex]\u00a0are used to describe the location of a point in space.<\/li>\n<li>The distance [latex]d[\/latex]\u00a0between points [latex](x_{1},y_{1},z_{1})[\/latex] and\u00a0[latex](x_{2},y_{2},z_{2})[\/latex]\u00a0is given by the formula [latex]d=\\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}[\/latex].<\/li>\n<li>In three dimensions, the equations [latex]x=a[\/latex],\u00a0[latex]y=b[\/latex], and\u00a0[latex]z=c[\/latex]\u00a0describe planes that are parallel to the coordinate planes.<\/li>\n<li>The standard equation of a sphere with center [latex](a,b,c)[\/latex] and radius [latex]r[\/latex]\u00a0is [latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex].<\/li>\n<li>In three dimensions, as in two, vectors are commonly expressed in component form, [latex]{\\bf{v}}=\\langle{x,y,z}\\rangle[\/latex], or in terms of the standard unit vectors,\u00a0[latex]x{\\bf{i}}+y{\\bf{j}}+z{\\bf{k}}[\/latex].<\/li>\n<li>Properties of vectors in space are a natural extension of the properties for vectors in a plane. Let\u00a0[latex]{\\bf{v}}=\\langle{x_{1},y_{1},z_{1}}\\rangle[\/latex] and\u00a0[latex]{\\bf{w}}=\\langle{x_{2},y_{2},z_{2}}\\rangle[\/latex]\u00a0be vectors, and let [latex]k[\/latex]\u00a0be a scalar.\n<ul>\n<li>Scalar multiplication:\u00a0[latex]k{\\bf{v}}=\\langle{kx_{1},ky_{1},kz_{1}}\\rangle[\/latex]<\/li>\n<li>Vector addition:\u00a0[latex]{\\bf{v}}+{\\bf{w}}=\\langle{x_{1},y_{1},z_{1}}\\rangle+\\langle{x_{2},y_{2},z_{2}}\\rangle=\\langle{x_{1}+x_{2},y_{1}+y_{2},z_{1}+z_{2}}\\rangle[\/latex]<\/li>\n<li>Vector subtraction:\u00a0[latex]{\\bf{v}}-{\\bf{w}}=\\langle{x_{1},y_{1},z_{1}}\\rangle-\\langle{x_{2},y_{2},z_{2}}\\rangle=\\langle{x_{1}-x_{2},y_{1}-y_{2},z_{1}-z_{2}}\\rangle[\/latex]<\/li>\n<li>Vector magnitude: [latex]\\parallel{\\bf{v}}\\parallel=\\sqrt{{x_{1}}^{2}+{y_{1}}^{2}+{z_{1}}^{2}}[\/latex]<\/li>\n<li>Unit vector in the direction of [latex]{\\bf{v}}[\/latex]: [latex]\\frac{\\bf{v}}{\\parallel{\\bf{v}}\\parallel}=\\frac{1}{\\parallel{\\bf{v}}\\parallel}\\langle{x_{1},y_{1},z_{1}}\\rangle=\\langle{\\frac{x_{1}}{\\parallel{\\bf{v}}\\parallel},\\frac{y_{1}}{\\parallel{\\bf{v}}\\parallel},\\frac{z_{1}}{\\parallel{\\bf{v}}\\parallel}}\\rangle,{\\bf{v}}\\ne{0}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/section>\n<p><strong>Vector Operations in Space<\/strong><\/p>\n<\/div>\n<ul>\n<li>The dot product, or scalar product, of two vectors\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex] and\u00a0[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex] is\u00a0[latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex].<\/li>\n<li>The dot product satisfies the following properties:\n<ul>\n<li>[latex]{\\bf{u}}\\cdot{\\bf{v}}={\\bf{v}}\\cdot{\\bf{u}}[\/latex]<\/li>\n<li>[latex]{\\bf{u}}\\cdot\\left({\\bf{v}}+{\\bf{w}}\\right)={\\bf{u}}\\cdot{\\bf{v}}+{\\bf{u}}\\cdot{\\bf{w}}[\/latex]<\/li>\n<li>[latex]c({\\bf{u}}\\cdot{\\bf{v}})=(c{\\bf{u}})\\cdot{\\bf{v}}={\\bf{u}}\\cdot(c{\\bf{v}})[\/latex]<\/li>\n<li>[latex]{\\bf{v}}\\cdot{\\bf{v}}=\\parallel{\\bf{v}}\\parallel^{2}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>The dot product of two vectors can be expressed, alternatively, as [latex]{\\bf{u}}\\cdot{\\bf{v}}=\\parallel{\\bf{u}}\\parallel\\parallel{\\bf{v}}\\parallel\\cos\\theta[\/latex].\u00a0This form of the dot product is useful for finding the measure of the angle formed by two vectors.<\/li>\n<li>Vectors\u00a0[latex]\\bf{u}[\/latex]\u00a0and\u00a0[latex]\\bf{v}[\/latex]\u00a0are orthogonal if [latex]{\\bf{u}}\\cdot{\\bf{v}}=0[\/latex].<\/li>\n<li>The angles formed by a nonzero vector and the coordinate axes are called the\u00a0<em data-effect=\"italics\">direction angles<\/em>\u00a0for the vector. The cosines of these angles are known as the\u00a0<em data-effect=\"italics\">direction cosines<\/em>.<\/li>\n<li>The vector projection of\u00a0[latex]\\bf{v}[\/latex]\u00a0onto\u00a0[latex]\\bf{u}[\/latex] is the vector [latex]\\text{proj}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel^{2}}{\\bf{u}}[\/latex].\u00a0The magnitude of this vector is known as the\u00a0<em data-effect=\"italics\">scalar projection<\/em>\u00a0of [latex]\\bf{v}[\/latex]\u00a0onto\u00a0[latex]\\bf{u}[\/latex], given by\u00a0[latex]\\text{comp}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel}[\/latex].<\/li>\n<li>\n<div class=\"os-section-area\">\n<section id=\"fs-id1163723268353\" class=\"key-concepts\" data-depth=\"1\">Work is done when a force is applied to an object, causing displacement. When the force is represented by the vector [latex]\\bf{F}[\/latex]\u00a0and the displacement is represented by the vector\u00a0[latex]\\bf{s}[\/latex], then the work done [latex]W[\/latex]\u00a0is given by the formula [latex]W={\\bf{F}}\\cdot{\\bf{s}}=\\parallel{\\bf{F}}\\parallel\\parallel{\\bf{s}}\\parallel\\cos\\theta[\/latex].<\/section>\n<\/div>\n<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Distance between two points in space<\/strong><br \/>\n[latex]d=\\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}[\/latex]<\/li>\n<li><strong>Sphere with center [latex](a,b,c)[\/latex] and radius [latex]r[\/latex]<\/strong><br \/>\n[latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex]<\/li>\n<li><strong>Dot product of u and v<br \/>\n<\/strong>[latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex]<\/li>\n<li><strong>Cosine of the angle formed by u and v<\/strong><br \/>\n[latex]\\cos\\theta=\\frac{{\\bf{u}}\\cdot{\\bf{v}}}{\\parallel{\\bf{u}}\\parallel\\parallel{\\bf{v}}\\parallel}[\/latex]<\/li>\n<li><strong>Vector projection of v onto u<\/strong><br \/>\n[latex]\\text{proj}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel^{2}}{\\bf{u}}[\/latex]<\/li>\n<li><strong>Scalar projection of v onto u<\/strong><br \/>\n[latex]\\text{comp}_{\\bf{u}}{\\bf{v}}=\\frac{\\bf{u}\\cdot\\bf{v}}{\\parallel{\\bf{u}}\\parallel}[\/latex]<\/li>\n<\/ul>\n<ul>\n<li><strong>Work done by a force F to move an object through displacement vector [latex]\\overrightarrow{PQ}[\/latex]<\/strong><br \/>\n[latex]W={\\bf{F}}\\cdot{\\bf{s}}=\\parallel{\\bf{F}}\\parallel\\parallel{\\bf{s}}\\parallel\\cos\\theta[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>component<\/dt>\n<dd>a scalar that describes either the vertical or horizontal direction of a vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>coordinate plane<\/dt>\n<dd>a plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: the [latex]xy[\/latex]-plane, [latex]xz[\/latex]-plane, or the [latex]yz[\/latex]<span style=\"font-size: 1em;\">-plane<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>direction angles<\/dt>\n<dd>the angles formed by a nonzero vector and the coordinate axes<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>direction cosines<\/dt>\n<dd>the cosines of the angles formed by a nonzero vector and the coordinate axes<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>dot product or scalar product<\/dt>\n<dd>[latex]{\\bf{u}}\\cdot{\\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[\/latex], where\u00a0[latex]{\\bf{u}}=\\langle{u_1,u_2,u_3}\\rangle[\/latex]\u00a0and[latex]{\\bf{v}}=\\langle{v_1,v_2,v_3}\\rangle[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>equivalent vectors<\/dt>\n<dd>vectors that have the same magnitude and the same direction<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>initial point<\/dt>\n<dd>the starting point of a vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>magnitude<\/dt>\n<dd>the length of a vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>normalization<\/dt>\n<dd>using scalar multiplication to find a unit vector with a given direction<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>octants<\/dt>\n<dd>the eight regions of space created by the coordinate planes<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>orthogonal vectors<\/dt>\n<dd>vectors that form a right angle when placed in standard position<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>parallelogram method<\/dt>\n<dd>a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of the vectors is the diagonal of that parallelogram<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>right-hand rule<\/dt>\n<dd>a common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the [latex]z[\/latex]<span style=\"font-size: 1em;\">-axis in such a way that the fingers curl from the positive [latex]x[\/latex]-axis to the positive [latex]y[\/latex]-axis, the thumb points in the direction of the positive [latex]z[\/latex]-axis<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>scalar<\/dt>\n<dd>a real number<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>scalar multiplication<\/dt>\n<dd>a vector operation that defines the product of a scalar and a vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>scalar projection<\/dt>\n<dd>the magnitude of the vector projection of a vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>sphere<\/dt>\n<dd>the set of all points equidistant from a given point known as the <em style=\"font-size: 1em;\" data-effect=\"italics\">center<\/em><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>standard equation of a sphere<\/dt>\n<dd>[latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[\/latex] describes a sphere with center [latex](a,b,c)[\/latex]<span class=\"Apple-converted-space\"><span style=\"color: #000000;\"><span style=\"caret-color: #000000; font-size: 14px; white-space: nowrap;\">\u00a0and radius [latex]r[\/latex]<\/span><\/span><\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>standard-position Vectors<\/dt>\n<dd>a vector with initial point [latex](0,0)[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>standard unit vectors<\/dt>\n<dd>unit vectors along the coordinate axes:\u00a0[latex]{\\bf{i}}=\\langle{1,0}\\rangle[\/latex], [latex]{\\bf{j}}=\\langle{0,1}\\rangle[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>terminal point<\/dt>\n<dd>the endpoint of a vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>three-dimensional rectangular coordinate system<\/dt>\n<dd>a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple\u00a0[latex](x,y,z)[\/latex]\u00a0that plots\u00a0its<span style=\"font-size: 1em;\">\u00a0location relative to the defining axes<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>triangle inequality<\/dt>\n<dd>the length of any side of a triangle is less than the sum of the lengths of the other two sides<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>triangle method<\/dt>\n<dd>a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of the first vector; the terminal point of the sum is the terminal point of the second vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>unit vector<\/dt>\n<dd>a vector with magnitude [latex]1[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector<\/dt>\n<dd>a mathematical object that has both magnitude and direction<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector addition<\/dt>\n<dd>a vector operation that defines the sum of two vectors<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector difference<\/dt>\n<dd>the vector difference\u00a0[latex]{\\bf{v}}-{\\bf{w}}[\/latex]\u00a0is defined as\u00a0[latex]{\\bf{v}}+(-{\\bf{w}})={\\bf{v}}+(-1){\\bf{w}}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector projection<\/dt>\n<dd>the component of a vector that follows a given direction<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>vector sum<\/dt>\n<dd>the sum of two vectors, [latex]{\\bf{v}}[\/latex] and [latex]{\\bf{w}}[\/latex]can be constructed graphically by placing the initial point of\u00a0[latex]{\\bf{w}}[\/latex]\u00a0at the terminal point of [latex]{\\bf{v}}[\/latex]; then the vector sum\u00a0[latex]{\\bf{v}}+{\\bf{w}}[\/latex]\u00a0is the vector with an initial point that coincides with the initial point of [latex]{\\bf{v}}[\/latex], and with a terminal point that coincides with the terminal point of\u00a0[latex]{\\bf{w}}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>work done by a force<\/dt>\n<dd>work is generally thought of as the amount of energy it takes to move an object; if we represent an applied force by a vector [latex]\\bf{F}[\/latex] and the displacement of an object by a vector [latex]\\bf{s}[\/latex], then the work done by the force is the dot product of [latex]\\bf{F}[\/latex]\u00a0and [latex]\\bf{s}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>zero vector<\/dt>\n<dd>the vector with both initial point and terminal point\u00a0[latex](0,0)[\/latex]<\/dd>\n<\/dl>\n","protected":false},"author":15,"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":[],"class_list":["post-486","chapter","type-chapter","status-publish","hentry"],"part":420,"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\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/486","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/486\/revisions"}],"predecessor-version":[{"id":899,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/486\/revisions\/899"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/420"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/486\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=486"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=486"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=486"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=486"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}