Gilbert Strang; Edwin “Jed” Herman

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 $v = 〈 x, y 〉 .$
The magnitude of a vector is a scalar: $‖ v ‖ = \sqrt{x^{2} + y^{2}} .$
A unit vector $u$ has magnitude $1$ and can be found by dividing a vector by its magnitude: $u = \frac{1}{‖ v ‖} v .$ The standard unit vectors are $i = 〈 1, 0 〉 and j = 〈 0, 1 〉 .$ A vector $v = 〈 x, y 〉$ can be expressed in terms of the standard unit vectors as $v = x i + y j .$
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 $(x, y, z)$ are used to describe the location of a point in space.
The distance $d$ between points $(x_{1}, y_{1}, z_{1})$ and $(x_{2}, y_{2}, z_{2})$ is given by the formula

$d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}} .$
In three dimensions, the equations $x = a, y = b, and z = c$ describe planes that are parallel to the coordinate planes.
The standard equation of a sphere with center $(a, b, c)$ and radius $r$ is

${(x - a)}^{2} + {(y - b)}^{2} + {(z - c)}^{2} = r^{2} .$
In three dimensions, as in two, vectors are commonly expressed in component form, $v = 〈 x, y, z 〉,$ or in terms of the standard unit vectors, $x i + y j + z k .$
Properties of vectors in space are a natural extension of the properties for vectors in a plane. Let v=〈x1,y1,z1〉v=〈x1,y1,z1〉 and w=〈x2,y2,z2〉w=〈x2,y2,z2〉 be vectors, and let kk be a scalar.
- Scalar multiplication: $k v = 〈 k x_{1}, k y_{1}, k z_{1} 〉$
- Vector addition: $v + w = 〈 x_{1}, y_{1}, z_{1} 〉 + 〈 x_{2}, y_{2}, z_{2} 〉 = 〈 x_{1} + x_{2}, y_{1} + y_{2}, z_{1} + z_{2} 〉$
- Vector subtraction: $v - w = 〈 x_{1}, y_{1}, z_{1} 〉 - 〈 x_{2}, y_{2}, z_{2} 〉 = 〈 x_{1} - x_{2}, y_{1} - y_{2}, z_{1} - z_{2} 〉$
- Vector magnitude: $‖ v ‖ = \sqrt{x_{1}^{2} + y_{1}^{2} + z_{1}^{2}}$
- Unit vector in the direction of v: $\frac{v}{‖ v ‖} = \frac{1}{‖ v ‖} 〈 x_{1}, y_{1}, z_{1} 〉 = 〈 \frac{x_{1}}{‖ v ‖}, \frac{y_{1}}{‖ v ‖}, \frac{z_{1}}{‖ v ‖} 〉,$ $v \neq 0$

2.3 The Dot Product

The dot product, or scalar product, of two vectors $u = 〈 u_{1}, u_{2}, u_{3} 〉$ and $v = 〈 v_{1}, v_{2}, v_{3} 〉$ is $u \cdot v = u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3} .$
The dot product satisfies the following properties:
- $u \cdot v = v \cdot u$
- $u \cdot (v + w) = u \cdot v + u \cdot w$
- $c (u \cdot v) = (c u) \cdot v = u \cdot (c v)$
- $v \cdot v = {‖ v ‖}^{2}$
The dot product of two vectors can be expressed, alternatively, as $u \cdot v = ‖ u ‖ ‖ v ‖ cos θ .$ 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 $u \cdot v = 0 .$
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 ${proj}_{u} v = \frac{u \cdot v}{{‖ u ‖}^{2}} u .$ The magnitude of this vector is known as the scalar projection of $v$ onto $u$ , given by ${comp}_{u} v = \frac{u \cdot v}{‖ u ‖} .$
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 $W = F \cdot s = ‖ F ‖ ‖ s ‖ cos θ .$

2.4 The Cross Product

The cross product $u \times v$ of two vectors $u = 〈 u_{1}, u_{2}, u_{3} 〉$ and $v = 〈 v_{1}, v_{2}, v_{3} 〉$ is a vector orthogonal to both $u$ and $v .$ Its length is given by $‖ u \times v ‖ = ‖ u ‖ \cdot ‖ v ‖ \cdot sin θ,$ where $θ$ is the angle between $u$ and $v .$ Its direction is given by the right-hand rule.
The algebraic formula for calculating the cross product of two vectors,
$u = 〈 u_{1}, u_{2}, u_{3} 〉 and v = 〈 v_{1}, v_{2}, v_{3} 〉,$ is
$u \times v = (u_{2} v_{3} - u_{3} v_{2}) i - (u_{1} v_{3} - u_{3} v_{1}) j + (u_{1} v_{2} - u_{2} v_{1}) k .$
The cross product satisfies the following properties for vectors u,v,andw,u,v,andw, and scalar c:c:
- $u \times v = − (v \times u)$
- $u \times (v + w) = u \times v + u \times w$
- $c (u \times v) = (c u) \times v = u \times (c v)$
- $u \times 0 = 0 \times u = 0$
- $v \times v = 0$
- $u \cdot (v \times w) = (u \times v) \cdot w$
The cross product of vectors $u = 〈 u_{1}, u_{2}, u_{3} 〉$ and $v = 〈 v_{1}, v_{2}, v_{3} 〉$ is the determinant $| \begin{matrix} i & j & k \\ u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \end{matrix} | .$
If vectors $u$ and $v$ form adjacent sides of a parallelogram, then the area of the parallelogram is given by $‖ u \times v ‖ .$
The triple scalar product of vectors $u,$ $v,$ and $w$ is $u \cdot (v \times w) .$
The volume of a parallelepiped with adjacent edges given by vectors $u, v, and w$ is $V = | u \cdot (v \times w) | .$
If the triple scalar product of vectors $u, v, and w$ 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 $τ$ measures the tendency of a force to produce rotation about an axis of rotation. If force $F$ is acting at a distance $r$ from the axis, then torque is equal to the cross product of $r$ and $F :$ $τ = r \times F .$

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 $v = 〈 a, b, c 〉$ passing through point $P = (x_{0}, y_{0}, z_{0})$ is $r = r_{0} + t v,$ where $r_{0} = 〈 x_{0}, y_{0}, z_{0} 〉$ is the position vector of point $P .$ This equation can be rewritten to form the parametric equations of the line: $x = x_{0} + t a,$ $y = y_{0} + t b,$ and $z = z_{0} + t c .$ 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 $v .$ If $Q$ is any point not on $L,$ then the distance from $Q$ to $L$ is $d = \frac{‖ \vec{P Q} \times v ‖}{‖ v ‖} .$
In three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew.
Given a point $P$ and vector $n,$ the set of all points $Q$ satisfying equation $n \cdot \vec{P Q} = 0$ forms a plane. Equation $n \cdot \vec{P Q} = 0$ is known as the vector equation of a plane.
The scalar equation of a plane containing point $P = (x_{0}, y_{0}, z_{0})$ with normal vector $n = 〈 a, b, c 〉$ is $a (x - x_{0}) + b (y - y_{0}) + c (z - z_{0}) = 0 .$ This equation can be expressed as $a x + b y + c z + d = 0,$ where $d = − a x_{0} - b y_{0} - c z_{0} .$ 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

$D = ‖ {proj}_{n} \vec{Q P} ‖ = | {comp}_{n} \vec{Q P} | = \frac{| \vec{Q P} \cdot n |}{‖ n ‖} .$
The normal vectors of parallel planes are parallel. When two planes intersect, they form a line.
The measure of the angle $θ$ between two intersecting planes can be found using the equation: $cos θ = \frac{| n_{1} \cdot n_{2} |}{‖ n_{1} ‖ ‖ n_{2} ‖},$ where $n_{1}$ and $n_{2}$ are normal vectors to the planes.
The distance $D$ from point $(x_{0}, y_{0}, z_{0})$ to plane $a x + b y + c z + d = 0$ is given by

$D = \frac{| a (x_{0} - x_{1}) + b (y_{0} - y_{1}) + c (z_{0} - z_{1}) |}{\sqrt{a^{2} + b^{2} + c^{2}}} = \frac{| a x_{0} + b y_{0} + c z_{0} + d |}{\sqrt{a^{2} + b^{2} + c^{2}}} .$

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 $z = 0, x = 0, or y = 0,$ 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 $A x^{2} + B y^{2} + C z^{2} + D x y + E x z + F y z + G x + H y + J z + K = 0 .$
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 $(r, θ, z),$ where $(r, θ)$ 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 $x = r cos θ,$ $y = r sin θ,$ and $z = z .$
To convert a point from Cartesian coordinates to cylindrical coordinates, use equations $r^{2} = x^{2} + y^{2},$ $tan θ = \frac{y}{x},$ and $z = z .$
In the spherical coordinate system, a point $P$ in space is represented by the ordered triple $(ρ, θ, φ),$ where $ρ$ is the distance between $P$ and the origin $(ρ \neq 0),$ $θ$ is the same angle used to describe the location in cylindrical coordinates, and $φ$ is the angle formed by the positive z-axis and line segment $\overset{—}{O P},$ where $O$ is the origin and $0 \leq φ \leq π .$
To convert a point from spherical coordinates to Cartesian coordinates, use equations $x = ρ sin φ cos θ,$ $y = ρ sin φ sin θ,$ and $z = ρ cos φ .$
To convert a point from Cartesian coordinates to spherical coordinates, use equations $ρ^{2} = x^{2} + y^{2} + z^{2},$ $tan θ = \frac{y}{x},$ and $φ = arccos (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}) .$
To convert a point from spherical coordinates to cylindrical coordinates, use equations $r = ρ sin φ,$ $θ = θ,$ and $z = ρ cos φ .$
To convert a point from cylindrical coordinates to spherical coordinates, use equations $ρ = \sqrt{r^{2} + z^{2}},$ $θ = θ,$ and $φ = arccos (\frac{z}{\sqrt{r^{2} + z^{2}}}) .$