### Key Concepts

### 2.1 Vectors in the Plane

- 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 $\text{v}=\langle x,y\rangle .$
- The magnitude of a vector is a scalar: $\Vert \text{v}\Vert =\sqrt{{x}^{2}+{y}^{2}}.$
- A unit vector $\text{u}$ has magnitude $1$ and can be found by dividing a vector by its magnitude: $\text{u}=\frac{1}{\Vert \text{v}\Vert}\text{v}.$ The standard unit vectors are $\text{i}=\langle 1,0\rangle \phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{j}=\langle 0,1\rangle .$ A vector $\text{v}=\langle x,y\rangle $ can be expressed in terms of the standard unit vectors as $\text{v}=x\text{i}+y\text{j}.$
- Vectors are often used in physics and engineering to represent forces and velocities, among other quantities.

### 2.2 Vectors in Three Dimensions

- The three-dimensional coordinate system is built around a set of three axes that intersect at right angles at a single point, the origin. Ordered triples $\left(x,y,z\right)$ are used to describe the location of a point in space.
- The distance $d$ between points $\left({x}_{1},{y}_{1},{z}_{1}\right)$ and $\left({x}_{2},{y}_{2},{z}_{2}\right)$ is given by the formula

$$d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}+{\left({z}_{2}-{z}_{1}\right)}^{2}}.$$ - In three dimensions, the equations $x=a,y=b,\text{and}\phantom{\rule{0.2em}{0ex}}z=c$ describe planes that are parallel to the coordinate planes.
- The standard equation of a sphere with center $\left(a,b,c\right)$ and radius $r$ is

$${\left(x-a\right)}^{2}+{\left(y-b\right)}^{2}+{\left(z-c\right)}^{2}={r}^{2}.$$ - In three dimensions, as in two, vectors are commonly expressed in component form, $\text{v}=\langle x,y,z\rangle ,$ or in terms of the standard unit vectors, $x\text{i}+y\text{j}+z\text{k}.$
- Properties of vectors in space are a natural extension of the properties for vectors in a plane. Let $\text{v}=\langle {x}_{1},{y}_{1},{z}_{1}\rangle $ and $\text{w}=\langle {x}_{2},{y}_{2},{z}_{2}\rangle $ be vectors, and let $k$ be a scalar.
**Scalar multiplication:**$k\text{v}=\langle k{x}_{1},k{y}_{1},k{z}_{1}\rangle $**Vector addition:**$\text{v}+\text{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 $**Vector subtraction:**$\text{v}-\text{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 $**Vector magnitude:**$\Vert \text{v}\Vert =\sqrt{{x}_{1}{}^{2}+{y}_{1}{}^{2}+{z}_{1}{}^{2}}$**Unit vector in the direction of v:**$\frac{\text{v}}{\Vert \text{v}\Vert}=\frac{1}{\Vert \text{v}\Vert}\langle {x}_{1},{y}_{1},{z}_{1}\rangle =\langle \frac{{x}_{1}}{\Vert \text{v}\Vert},\frac{{y}_{1}}{\Vert \text{v}\Vert},\frac{{z}_{1}}{\Vert \text{v}\Vert}\rangle ,$ $\text{v}\ne 0$

### 2.3 The Dot Product

- The dot product, or scalar product, of two vectors $\text{u}=\langle {u}_{1},{u}_{2},{u}_{3}\rangle $ and $\text{v}=\langle {v}_{1},{v}_{2},{v}_{3}\rangle $ is $\text{u}\xb7\text{v}={u}_{1}{v}_{1}+{u}_{2}{v}_{2}+{u}_{3}{v}_{3}.$
- The dot product satisfies the following properties:
- $\text{u}\xb7\text{v}=\text{v}\xb7\text{u}$
- $\text{u}\xb7\left(\text{v}+\text{w}\right)=\text{u}\xb7\text{v}+\text{u}\xb7\text{w}$
- $c\left(\text{u}\xb7\text{v}\right)=\left(c\text{u}\right)\xb7\text{v}=\text{u}\xb7\left(c\text{v}\right)$
- $\text{v}\xb7\text{v}={\Vert \text{v}\Vert}^{2}$

- The dot product of two vectors can be expressed, alternatively, as $\text{u}\xb7\text{v}=\Vert \text{u}\Vert \Vert \text{v}\Vert \text{cos}\phantom{\rule{0.2em}{0ex}}\theta .$ This form of the dot product is useful for finding the measure of the angle formed by two vectors.
- Vectors
**u**and**v**are orthogonal if $\text{u}\xb7\text{v}=0.$ - 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
**v**onto**u**is the vector ${\text{proj}}_{\text{u}}\text{v}=\frac{\text{u}\xb7\text{v}}{{\Vert \text{u}\Vert}^{2}}\text{u}.$ The magnitude of this vector is known as the*scalar projection*of**v**onto**u**, given by ${\text{comp}}_{\text{u}}\text{v}=\frac{\text{u}\xb7\text{v}}{\Vert \text{u}\Vert}.$ - Work is done when a force is applied to an object, causing displacement. When the force is represented by the vector
**F**and the displacement is represented by the vector**s**, then the work done*W*is given by the formula $W=\text{F}\xb7s=\Vert \text{F}\Vert \Vert \text{s}\Vert \text{cos}\phantom{\rule{0.2em}{0ex}}\theta .$

### 2.4 The Cross Product

- The cross product $\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}$ of two vectors $\text{u}=\langle {u}_{1},{u}_{2},{u}_{3}\rangle $ and $\text{v}=\langle {v}_{1},{v}_{2},{v}_{3}\rangle $ is a vector orthogonal to both $\text{u}$ and $\text{v}.$ Its length is given by $\Vert \text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}\Vert =\Vert \text{u}\Vert \xb7\Vert \text{v}\Vert \xb7\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ,$ where $\theta $ is the angle between $\text{u}$ and $\text{v}.$ Its direction is given by the right-hand rule.
- The algebraic formula for calculating the cross product of two vectors,

$\text{u}=\langle {u}_{1},{u}_{2},{u}_{3}\rangle \phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{v}=\langle {v}_{1},{v}_{2},{v}_{3}\rangle ,$ is

$\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}=\left({u}_{2}{v}_{3}-{u}_{3}{v}_{2}\right)\text{i}-\left({u}_{1}{v}_{3}-{u}_{3}{v}_{1}\right)\text{j}+\left({u}_{1}{v}_{2}-{u}_{2}{v}_{1}\right)\text{k}.$ - The cross product satisfies the following properties for vectors $\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w},$ and scalar $c\text{:}$
- $\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}=\text{\u2212}\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{u}\right)$
- $\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\left(\text{v}+\text{w}\right)=\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}+\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{w}$
- $c\left(\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}\right)=\left(c\text{u}\right)\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}=\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\left(c\text{v}\right)$
- $\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}0=0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{u}=0$
- $\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}=0$
- $\text{u}\xb7\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{w}\right)=\left(\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}\right)\xb7\text{w}$

- The cross product of vectors $\text{u}=\langle {u}_{1},{u}_{2},{u}_{3}\rangle $ and $\text{v}=\langle {v}_{1},{v}_{2},{v}_{3}\rangle $ is the determinant $\left|\begin{array}{ccc}\text{i}\hfill & \text{j}\hfill & \text{k}\hfill \\ {u}_{1}\hfill & {u}_{2}\hfill & {u}_{3}\hfill \\ {v}_{1}\hfill & {v}_{2}\hfill & {v}_{3}\hfill \end{array}\right|.$
- If vectors $\text{u}$ and $\text{v}$ form adjacent sides of a parallelogram, then the area of the parallelogram is given by $\Vert \text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}\Vert .$
- The triple scalar product of vectors $\text{u},$ $\text{v},$ and $\text{w}$ is $\text{u}\xb7\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{w}\right).$
- The volume of a parallelepiped with adjacent edges given by vectors $\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ is $V=\left|\text{u}\xb7\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{w}\right)\right|.$
- If the triple scalar product of vectors $\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ is zero, then the vectors are coplanar. The converse is also true: If the vectors are coplanar, then their triple scalar product is zero.
- The cross product can be used to identify a vector orthogonal to two given vectors or to a plane.
- Torque $\tau $ measures the tendency of a force to produce rotation about an axis of rotation. If force $\text{F}$ is acting at a distance $\text{r}$ from the axis, then torque is equal to the cross product of $\text{r}$ and $\text{F}\text{:}$ $\tau =\text{r}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{F}.$

### 2.5 Equations of Lines and Planes in Space

- In three dimensions, the direction of a line is described by a direction vector. The vector equation of a line with direction vector $\text{v}=\langle a,b,c\rangle $ passing through point $P=\left({x}_{0},{y}_{0},{z}_{0}\right)$ is $\text{r}={\text{r}}_{0}+t\text{v},$ where ${\text{r}}_{0}=\langle {x}_{0},{y}_{0},{z}_{0}\rangle $ is the position vector of point $P.$ This equation can be rewritten to form the parametric equations of the line: $x={x}_{0}+ta,$ $y={y}_{0}+tb,$ and $z={z}_{0}+tc.$ The line can also be described with the symmetric equations $\frac{x-{x}_{0}}{a}=\frac{y-{y}_{0}}{b}=\frac{z-{z}_{0}}{c}.$
- Let $L$ be a line in space passing through point $P$ with direction vector $\text{v}.$ If $Q$ is any point not on $L,$ then the distance from $Q$ to $L$ is $d=\frac{\Vert \overrightarrow{PQ}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}\Vert}{\Vert \text{v}\Vert}.$
- In three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew.
- Given a point $P$ and vector $\text{n},$ the set of all points $Q$ satisfying equation $\text{n}\xb7\overrightarrow{PQ}=0$ forms a plane. Equation $\text{n}\xb7\overrightarrow{PQ}=0$ is known as the
*vector equation of a plane.* - The scalar equation of a plane containing point $P=\left({x}_{0},{y}_{0},{z}_{0}\right)$ with normal vector $\text{n}=\langle a,b,c\rangle $ is $a\left(x-{x}_{0}\right)+b\left(y-{y}_{0}\right)+c\left(z-{z}_{0}\right)=0.$ This equation can be expressed as $ax+by+cz+d=0,$ where $d=\text{\u2212}a{x}_{0}-b{y}_{0}-c{z}_{0}.$ This form of the equation is sometimes called the
*general form of the equation of a plane*. - Suppose a plane with normal vector
**n**passes through point $Q.$ The distance $D$ from the plane to point $P$ not in the plane is given by

$$D=\Vert {\text{proj}}_{\text{n}}\overrightarrow{QP}\Vert =\left|{\text{comp}}_{\text{n}}\overrightarrow{QP}\right|=\frac{\left|\overrightarrow{QP}\xb7\text{n}\right|}{\Vert \text{n}\Vert}.$$ - The normal vectors of parallel planes are parallel. When two planes intersect, they form a line.
- The measure of the angle $\theta $ between two intersecting planes can be found using the equation: $\text{cos}\phantom{\rule{0.2em}{0ex}}\theta =\frac{\left|{\text{n}}_{1}\xb7{\text{n}}_{2}\right|}{\Vert {\text{n}}_{1}\Vert \Vert {\text{n}}_{2}\Vert},$ where ${\text{n}}_{1}$ and ${\text{n}}_{2}$ are normal vectors to the planes.
- The distance $D$ from point $\left({x}_{0},{y}_{0},{z}_{0}\right)$ to plane $ax+by+cz+d=0$ is given by

$$D=\frac{\left|a\left({x}_{0}-{x}_{1}\right)+b\left({y}_{0}-{y}_{1}\right)+c\left({z}_{0}-{z}_{1}\right)\right|}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}=\frac{\left|a{x}_{0}+b{y}_{0}+c{z}_{0}+d\right|}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}.$$

### 2.6 Quadric Surfaces

- A set of lines parallel to a given line passing through a given curve is called a
*cylinder*, or a*cylindrical surface*. The parallel lines are called*rulings*. - The intersection of a three-dimensional surface and a plane is called a
*trace*. To find the trace in the*xy*-,*yz*-, or*xz*-planes, set $z=0,x=0,\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}y=0,$ respectively. - Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the form $A{x}^{2}+B{y}^{2}+C{z}^{2}+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.$
- To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.
- Important quadric surfaces are summarized in Figure 2.87 and Figure 2.88.

### 2.7 Cylindrical and Spherical Coordinates

- In the cylindrical coordinate system, a point in space is represented by the ordered triple $\left(r,\theta ,z\right),$ where $\left(r,\theta \right)$ represents the polar coordinates of the point’s projection in the
*xy*-plane and $z$ represents the point’s projection onto the*z*-axis. - To convert a point from cylindrical coordinates to Cartesian coordinates, use equations $x=r\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,$ $y=r\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ,$ and $z=z.$
- To convert a point from Cartesian coordinates to cylindrical coordinates, use equations ${r}^{2}={x}^{2}+{y}^{2},$ $\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =\frac{y}{x},$ and $z=z.$
- In the spherical coordinate system, a point $P$ in space is represented by the ordered triple $\left(\rho ,\theta ,\phi \right),$ where $\rho $ is the distance between $P$ and the origin $\left(\rho \ne 0\right),$ $\theta $ is the same angle used to describe the location in cylindrical coordinates, and $\phi $ is the angle formed by the positive
*z*-axis and line segment $\stackrel{\u2014}{OP},$ where $O$ is the origin and $0\le \phi \le \pi .$ - To convert a point from spherical coordinates to Cartesian coordinates, use equations $x=\rho \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\phi \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,$ $y=\rho \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\phi \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ,$ and $z=\rho \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\phi .$
- To convert a point from Cartesian coordinates to spherical coordinates, use equations ${\rho}^{2}={x}^{2}+{y}^{2}+{z}^{2},$ $\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =\frac{y}{x},$ and $\phi =\text{arccos}\left(\frac{z}{\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}}\right).$
- To convert a point from spherical coordinates to cylindrical coordinates, use equations $r=\rho \phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\phi ,$ $\theta =\theta ,$ and $z=\rho \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\phi .$
- To convert a point from cylindrical coordinates to spherical coordinates, use equations $\rho =\sqrt{{r}^{2}+{z}^{2}},$ $\theta =\theta ,$ and $\phi =\text{arccos}\left(\frac{z}{\sqrt{{r}^{2}+{z}^{2}}}\right).$