Essential Concepts

Vector Basics

• Vectors are used to represent quantities that have both magnitude and direction.
• 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.
• Subtraction of vectors is defined in terms of adding the negative of the vector.
• A vector is written in component form as [latex]{\bf{v}}=\langle{x,y}\rangle[/latex].
• The magnitude of a vector is a scalar: [latex]\parallel{\bf{v}}\parallel=\sqrt{x^2+y^2}[/latex].
• A unit vector [latex]\bf{u}[/latex] has magnitude [latex]1[/latex] and can be found by dividing a vector by its magnitude: [latex]{\bf{u}}=\frac{1}{\parallel{\bf{v}}\parallel}{\bf{v}}[/latex]  The standard unit vectors are [latex]{\bf{i}}=\langle{1,0}\rangle[/latex] and [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 [latex]{\bf{v}}=x{\bf{i}}+y{\bf{j}}[/latex].

• The dot product, or scalar product, of two vectors [latex]{\bf{u}}=\langle{u_1,u_2,u_3}\rangle[/latex] and [latex]{\bf{v}}=\langle{v_1,v_2,v_3}\rangle[/latex] is [latex]{\bf{u}}\cdot{\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[/latex].
• The dot product satisfies the following properties:
  • [latex]{\bf{u}}\cdot{\bf{v}}={\bf{v}}\cdot{\bf{u}}[/latex]
  • [latex]{\bf{u}}\cdot\left({\bf{v}}+{\bf{w}}\right)={\bf{u}}\cdot{\bf{v}}+{\bf{u}}\cdot{\bf{w}}[/latex]
  • [latex]c({\bf{u}}\cdot{\bf{v}})=(c{\bf{u}})\cdot{\bf{v}}={\bf{u}}\cdot(c{\bf{v}})[/latex]
  • [latex]{\bf{v}}\cdot{\bf{v}}=\parallel{\bf{v}}\parallel^{2}[/latex]
• 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]. This form of the dot product is useful for finding the measure of the angle formed by two vectors.
• Vectors [latex]\bf{u}[/latex] and [latex]\bf{v}[/latex] are orthogonal if [latex]{\bf{u}}\cdot{\bf{v}}=0[/latex].
• The angles formed by a nonzero vector and the coordinate axes are called the direction angles for the vector. The cosines of these angles are known as the direction cosines.
• The vector projection of [latex]\bf{v}[/latex] onto [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]. The magnitude of this vector is known as the scalar projection of [latex]\bf{v}[/latex] onto [latex]\bf{u}[/latex], given by [latex]\text{comp}_{\bf{u}}{\bf{v}}=\frac{\bf{u}\cdot\bf{v}}{\parallel{\bf{u}}\parallel}[/latex].
• 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] and the displacement is represented by the vector [latex]\bf{s}[/latex], then the work done [latex]W[/latex] is given by the formula [latex]W={\bf{F}}\cdot{\bf{s}}=\parallel{\bf{F}}\parallel\parallel{\bf{s}}\parallel\cos\theta[/latex].

Key Equations

• Distance between two points in space
  [latex]d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}[/latex]
• Sphere with center [latex](a,b,c)[/latex] and radius [latex]r[/latex]
  [latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[/latex]
• Dot product of u and v
  [latex]{\bf{u}}\cdot{\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[/latex]
• Cosine of the angle formed by u and v
  [latex]\cos\theta=\frac{{\bf{u}}\cdot{\bf{v}}}{\parallel{\bf{u}}\parallel\parallel{\bf{v}}\parallel}[/latex]
• Vector projection of v onto u
  [latex]\text{proj}_{\bf{u}}{\bf{v}}=\frac{\bf{u}\cdot\bf{v}}{\parallel{\bf{u}}\parallel^{2}}{\bf{u}}[/latex]
• Scalar projection of v onto u
  [latex]\text{comp}_{\bf{u}}{\bf{v}}=\frac{\bf{u}\cdot\bf{v}}{\parallel{\bf{u}}\parallel}[/latex]

• Work done by a force F to move an object through displacement vector [latex]\overrightarrow{PQ}[/latex]
  [latex]W={\bf{F}}\cdot{\bf{s}}=\parallel{\bf{F}}\parallel\parallel{\bf{s}}\parallel\cos\theta[/latex]

Glossary

component

a scalar that describes either the vertical or horizontal direction of a vector

coordinate plane

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]-plane

direction angles

the angles formed by a nonzero vector and the coordinate axes

direction cosines

the cosines of the angles formed by a nonzero vector and the coordinate axes

dot product or scalar product

[latex]{\bf{u}}\cdot{\bf{v}}=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}[/latex], where [latex]{\bf{u}}=\langle{u_1,u_2,u_3}\rangle[/latex] and[latex]{\bf{v}}=\langle{v_1,v_2,v_3}\rangle[/latex]

equivalent vectors

vectors that have the same magnitude and the same direction

initial point

the starting point of a vector

magnitude

the length of a vector

normalization

using scalar multiplication to find a unit vector with a given direction

octants

the eight regions of space created by the coordinate planes

orthogonal vectors

vectors that form a right angle when placed in standard position

parallelogram method

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

right-hand rule

a common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the [latex]z[/latex]-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

scalar

a real number

scalar multiplication

a vector operation that defines the product of a scalar and a vector

scalar projection

the magnitude of the vector projection of a vector

sphere

the set of all points equidistant from a given point known as the center

standard equation of a sphere

[latex](x-a)^{2}+(y-b)^{2}+(z-c)^{2}=r^{2}[/latex] describes a sphere with center [latex](a,b,c)[/latex] and radius [latex]r[/latex]

standard-position Vectors

a vector with initial point [latex](0,0)[/latex]

standard unit vectors

unit vectors along the coordinate axes: [latex]{\bf{i}}=\langle{1,0}\rangle[/latex], [latex]{\bf{j}}=\langle{0,1}\rangle[/latex]

terminal point

the endpoint of a vector

three-dimensional rectangular coordinate system

a coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple [latex](x,y,z)[/latex] that plots its location relative to the defining axes

triangle inequality

the length of any side of a triangle is less than the sum of the lengths of the other two sides

triangle method

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

unit vector

a vector with magnitude [latex]1[/latex]

vector

a mathematical object that has both magnitude and direction

vector addition

a vector operation that defines the sum of two vectors

vector difference

the vector difference [latex]{\bf{v}}-{\bf{w}}[/latex] is defined as [latex]{\bf{v}}+(-{\bf{w}})={\bf{v}}+(-1){\bf{w}}[/latex]

vector projection

the component of a vector that follows a given direction

vector sum

the sum of two vectors, [latex]{\bf{v}}[/latex] and [latex]{\bf{w}}[/latex]can be constructed graphically by placing the initial point of [latex]{\bf{w}}[/latex] at the terminal point of [latex]{\bf{v}}[/latex]; then the vector sum [latex]{\bf{v}}+{\bf{w}}[/latex] is 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 [latex]{\bf{w}}[/latex]

work done by a force

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] and [latex]\bf{s}[/latex]

zero vector

the vector with both initial point and terminal point [latex](0,0)[/latex]