This toolkit contains all of the definitions, properties, facts and theorems you need to memorize all on one webpage. These are your tools on quizzes and exams.
Table of Contents
A set is a collection of objects, along with a condition of membership. We write \[A = \{\text{objects} : \text{membership condition}\}\]
If $a$ is an element of the set $S$, we write $a \in S$.
$\mathbb{R}$ denotes the set of all real numbers. The familiar $xy$-plane is \[\mathbb{R}^2 = \{(x, y) : x, y \in \mathbb{R}\}\]
The 3D rectangular coordinate system is \[\mathbb{R}^3 = \{(x, y, z) : x, y, z \in \mathbb{R}\}\]
Movement in each coordinate $a, b, c$ of the point $(a, b, c)$ is parallel to the $x$-, $y$-, and $z$-axis, respectively.
There are two directions the positive $z$-axis could point. Mathematicians decided on the right hand rule: flatten your hand and point your four fingers along the $x$-axis. Curl your fingers in the shortest rotation from the $x$-axis to the $y$-axis. Then your thumb points in the direction of the positive $z$-axis.
The distance between the points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$, denoted $|P_1P_2|$, is \[|P_1P_2| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
A sphere is the set of all points $P(x, y, z)$ at distance $r$ away from the center $C(h, k, \ell)$. Its equation is \[(x - h)^2 + (y - k)^2 + (z - \ell)^2 = r^2\]
If the center is $O(0,0,0)$, then the equation is \[x^2 + y^2 + z^2 = r^2\]
A vector is a mathematical quantity with both direction and magnitude. Vectors carry no information about position.
A displacement vector $\ray{AB}$ is a vector that takes you from a point $A$ to a point $B$.
Two vectors are equivalent (we write $\ray{AB} = \ray{CD}$) if they have the same direction and magnitude. Their initial points may be different, yet the vectors are equivalent.
The zero vector, denoted $\vec{0}$, is the vector with magnitude $0$. It is the one exception to the definition above: it has no assigned direction.
Given two vectors $\vec{u}$ and $\vec{v}$, position the tail of $\vec{v}$ on the tip of $\vec{u}$. Then $\vec{u} + \vec{v}$ is the vector from the tail of $\vec{u}$ to the tip of $\vec{v}$.
Let $c$ be a scalar and $\vec{v}$ a vector. The scalar multiple $c\vec{v}$ is the vector whose magnitude is $|c|$ times the magnitude of $\vec{v}$ and whose direction is the same as $\vec{v}$ if $c > 0$ and opposite to $\vec{v}$ if $c < 0$.
If $c = 0$ or $\vec{v} = \vec{0}$, then $c\vec{v} = \vec{0}$.
Scaling changes magnitude and either preserves or reverses direction. Every scalar multiple of $\vec{v}$ lies along the same line. Collecting all of them at once, the set \[\left\{c\,\vec{v} : c \in \mathbb{R}\right\}\] is a line in the direction of $\vec{v}$.
The difference of two vectors is defined as \[\vec{u} - \vec{v} = \vec{u} + (-\vec{v})\]
Think Reverse the direction of $\vec{v}$, then add normally.
If the initial points of both vectors are the same, $\vec{u} - \vec{v}$ is simply the vector from the tip of $\vec{v}$ to the tip of $\vec{u}$.
If we let the initial point of a vector $\vec{a}$ be the origin, then the terminal point has coordinates $(a_1, a_2)$ or $(a_1, a_2, a_3)$, and we write \[\vec{a} = \vc{a_1, a_2} \qquad \text{or} \qquad \vec{a} = \vc{a_1, a_2, a_3}\]
Angled brackets distinguish a vector from a coordinate $(a_1, a_2)$. These vectors are called position vectors.
Given points $A(a_1, a_2, a_3)$ and $B(b_1, b_2, b_3)$, the (position) vector $\vec{v}$ with representation $\ray{AB}$ is \[\vec{v} = \vc{b_1 - a_1,\ b_2 - a_2,\ b_3 - a_3}\] A vector is a difference of points (destination minus start). This is why a vector has no information about position: the difference forgets where the vector started.
Let $\vec{a} = \vc{a_1, a_2, a_3}$, $\vec{b} = \vc{b_1, b_2, b_3}$ and $c \in \mathbb{R}$.
The set of all $n$-dimensional vectors is \[V_n = \left\{\vc{a_1, a_2, \dots, a_n} : a_i \in \mathbb{R}\right\}\]
$V_n$ is called a vector space: a set of vectors, together with addition and scalar multiplication, required to satisfy the 8 properties below (the vector space axioms, for those in Calculus IV).
Let $\vec{a}, \vec{b}, \vec{c} \in V_n$ and $c, d \in \mathbb{R}$.
A unit vector is a vector with magnitude 1. Given $\vec{a} \neq \vec{0}$, the corresponding unit vector is \[\vec{u} = \dfrac{\vec{a}}{|\vec{a}|}\]
Any vector $\vec{a} \neq \vec{0}$ can be decomposed into a product of its length and its unit direction: \[\vec{a} = \underbrace{\abs{\vec{a}}}_{\text{length}} \cdot \underbrace{\dfrac{\vec{a}}{|\vec{a}|}}_{\text{unit direction}}\]
An arbitrary vector $\vec{a} = \vc{a_1, a_2, a_3}$ can be broken up into vectors in each dimension: \begin{align} \vec{a} &= \vc{a_1, a_2, a_3} \\ &= \vc{a_1, 0, 0} + \vc{0, a_2, 0} + \vc{0, 0, a_3} \\ &= a_1\vc{1,0,0} + a_2\vc{0,1,0} + a_3\vc{0,0,1} \\ &= a_1\iv + a_2\jv + a_3\kv \end{align} where $\iv = \vc{1,0,0}$, $\jv = \vc{0,1,0}$, $\kv = \vc{0,0,1}$ are the standard basis vectors.
Given two vectors $\vec{a} = \vc{a_1, a_2, a_3}$ and $\vec{b} = \vc{b_1, b_2, b_3}$, the dot product $\vec{a}\cdot\vec{b}$ is defined \[\vec{a}\cdot\vec{b} = a_1b_1 + a_2b_2 + a_3b_3\]
Let $\vec{a}, \vec{b}, \vec{c} \in V_3$ and $c \in \mathbb{R}$. Then
If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then \[\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta\]
Key Idea $\vec{a}\cdot\vec{b}$ is the Pythagorean defect: how far off $\vec{a}$ and $\vec{b}$ are from forming a right angle.
If $\vec{a}, \vec{b} \neq \vec{0}$, then the angle $\theta$ between $\vec{a}$ and $\vec{b}$ satisfies \[\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}\]
Two vectors are orthogonal (perpendicular) precisely when the angle between them is $\theta = 90^\circ$.
Fact $\vec{a}$ and $\vec{b}$ are orthogonal if and only if $\vec{a}\cdot\vec{b} = 0$.
The angle $\theta$, and in turn the dot product, tells us how two nonzero vectors are oriented:
Vectors are parallel when $\theta = 0^\circ$ or $180^\circ$, or equivalently when $\vec{b} = c\vec{a}$ for some scalar $c$.
$\vec{a}\cdot\vec{b}$ really measures "how much" $\vec{b}$ is pointing in $\vec{a}$'s direction, relative to $|\vec{a}|$.
When $|\vec{a}| = 1$, $\vec{a}\cdot\vec{b}$ answers the question "How much of $\vec{b}$ points along $\vec{a}$, measured in $\vec{a}$-lengths?"
In general, $\vec{a}$ is not a unit vector, but \[\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta \qquad \text{implies} \qquad |\vec{b}|\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|} = \dfrac{\vec{a}}{|\vec{a}|}\cdot \vec{b}\] which is exactly the dot product of $\vec{b}$ with the unit vector in the direction of $\vec{a}$.
The scalar projection of $\vec{b}$ onto $\vec{a}$ (also called the component of $\vec{b}$ along $\vec{a}$) is the signed length of the shadow of $\vec{b}$ in the direction of $\vec{a}$: \[\text{comp}_{\vec{a}}\vec{b} = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|}\]
The vector projection of $\vec{b}$ onto $\vec{a}$ is that shadow as a vector (scalar projection times the unit vector in the direction of $\vec{a}$): \[\text{proj}_{\vec{a}}\vec{b} = \left(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|}\right)\dfrac{\vec{a}}{|\vec{a}|} = \left(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|^2}\right)\vec{a}\]
If $A, B, C$ are distinct points on circle $O$ where $\overline{AB}$ is a diameter, then $\angle ACB$ is a right angle.
If $A(x_1, y_1)$ and $B(x_2, y_2)$ are the endpoints of a diameter, then the circle is the set of points $C(x, y)$ with $\ray{CA}\cdot\ray{CB} = 0$, giving \[(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0\]
A determinant of order 2 is defined by \[\begin{vmatrix} a & b \\ c & d\end{vmatrix} = ad - bc\]
Insert the coordinates of $\vec{a}$ and $\vec{b}$ into the determinant column-wise, $\vec{a}$ first. The result is the signed area of the parallelogram determined by $\vec{a}$ and $\vec{b}$.
The sign comes from the right hand rule, with the left column vector rotated onto the right column vector:
Swapping the order of the columns (rotating $\vec{b}$ onto $\vec{a}$ instead) flips the sign.
Let $\vec{a} = \vc{a_1, a_2, a_3}$ and $\vec{b} = \vc{b_1, b_2, b_3}$. The cross product of $\vec{a}$ and $\vec{b}$ is the vector \[\vec{a}\times\vec{b} = \vc{a_2b_3 - a_3b_2,\ a_3b_1 - a_1b_3,\ a_1b_2 - a_2b_1}\]
A determinant of order 3 is defined by expansion along the first row: \[\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{vmatrix} = a_1\begin{vmatrix} b_2 & b_3 \\ c_2 & c_3\end{vmatrix} - a_2\begin{vmatrix} b_1 & b_3 \\ c_1 & c_3\end{vmatrix} + a_3\begin{vmatrix} b_1 & b_2 \\ c_1 & c_2\end{vmatrix}\]
Note the alternating signs $+, -, +$.Let $\vec{a} = \vc{a_1, a_2, a_3}$ and $\vec{b} = \vc{b_1, b_2, b_3}$. The cross product $\vec{a}\times\vec{b}$ is \[\vec{a}\times\vec{b} = \begin{vmatrix} \iv & \jv & \kv \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{vmatrix} = \begin{vmatrix} a_2 & a_3 \\ b_2 & b_3\end{vmatrix}\iv - \begin{vmatrix} a_1 & a_3 \\ b_1 & b_3\end{vmatrix}\jv + \begin{vmatrix} a_1 & a_2 \\ b_1 & b_2\end{vmatrix}\kv\]
The vector $\vec{a}\times\vec{b}$ is orthogonal to both $\vec{a}$ and $\vec{b}$: \[(\vec{a}\times\vec{b})\cdot\vec{a} = 0 \qquad \text{and} \qquad (\vec{a}\times\vec{b})\cdot\vec{b} = 0\]
Among the two directions orthogonal to both, the direction of $\vec{a}\times\vec{b}$ is given by the right hand rule.
Careful Always curl your fingers from the vector left of $\times$ towards the vector on the right.
A vector $\vec{a}$ that is orthogonal to every vector in the plane.
If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ with $\theta \in [0, \pi]$, then \[|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta\]
The length of the cross product $\vec{a}\times\vec{b}$ is equal to the area of the parallelogram determined by $\vec{a}$ and $\vec{b}$.
Let $\vec{a}, \vec{b} \in V_3$ be nonzero. Then \[\vec{a}\times\vec{b} = \vc{A_{yz},\ A_{zx},\ A_{xy}}\] where $A_{yz}$ is the signed area of the parallelogram determined by $\vec{a}, \vec{b}$ after projecting onto the $yz$-plane, and likewise for $A_{zx}$ and $A_{xy}$.
Given $\vec{a}\times\vec{b}$, after you apply the right hand rule, look down the positive axis normal to that plane (toward the origin).
Projecting a surface onto a coordinate plane: set the coordinate along the plane's normal to 0.
Let $\vec{a}, \vec{b}, \vec{c} \in V_3$ and $c \in \mathbb{R}$. Then
In particular, $\vec{a}\times\vec{a} = \vec{0}$ for every $\vec{a} \in V_3$.
The product $\vec{a}\cdot(\vec{b}\times\vec{c})$ is called the scalar triple product.\[\vec{a}\cdot(\vec{b}\times\vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3\end{vmatrix}\]
The volume of the parallelepiped determined by the vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is the absolute value of their scalar triple product: \[V = \left|\vec{a}\cdot(\vec{b}\times\vec{c})\right|\]
In particular, $\vec{a}, \vec{b}, \vec{c}$ lie in the same plane (are coplanar) if and only if $\vec{a}\cdot(\vec{b}\times\vec{c}) = 0$.
A line $L$ in $\mathbb{R}^3$ is completely determined if we know two things: a point on the line and a direction vector for $L$, which is a vector $\vec{v}$ parallel to the line.
The vector equation of a line through a point $P_0(x_0, y_0, z_0)$ with position vector $\vec{r}_0$ and parallel to the vector $\vec{v}$ is \[\vec{r} = \vec{r}_0 + t\vec{v}, \qquad t \in \mathbb{R}\]
Letting $\vec{v} = \vc{a, b, c}$, $\vec{r} = \vc{x, y, z}$ and $\vec{r}_0 = \vc{x_0, y_0, z_0}$, this becomes \[\vc{x, y, z} = \vc{x_0 + ta,\ y_0 + tb,\ z_0 + tc}\]
Parametric equations for a line through the point $(x_0, y_0, z_0)$ and parallel to the vector $\vec{v} = \vc{a, b, c}$ are \[x = x_0 + at, \qquad y = y_0 + bt, \qquad z = z_0 + ct\]
The line segment from $\vec{r}_0$ to $\vec{r}_1$ is given by the vector equation \[\vec{r}(t) = (1 - t)\vec{r}_0 + t\vec{r}_1, \qquad 0 \leq t \leq 1\]
A plane in $\mathbb{R}^3$ is completely determined if we know two things:
The vector equation of a plane through a point $P_0(x_0, y_0, z_0)$ with position vector $\vec{r}_0$, normal vector $\vec{n}$, and arbitrary point $P(x, y, z)$ with position vector $\vec{r}$ is \[\vec{n}\cdot\left(\vec{r} - \vec{r}_0\right) = 0\] $\vec{r} - \vec{r}_0$ sweeps out every point in the plane.
A scalar equation of the plane through the point $P_0(x_0, y_0, z_0)$ with normal vector $\vec{n} = \vc{a, b, c}$ is \[a(x - x_0) + b(y - y_0) + c(z - z_0) = 0\]
Think of parameter $t$ as time, and $(x(t), y(t))$ as the location of a moving point. This allows us to trace curves which do not pass the vertical line test.
Vector $\vc{x(t), y(t)}$ is a position vector with tip tracing out the curve.
A conic section is an intersection of a double cone and a plane in $\mathbb{R}^3$.
| Conic | Rectangular Equation | Parametric Equations |
|---|---|---|
| Circle | \( (x-h)^2 + (y-k)^2 = r^2 \) | \( \begin{aligned} x &= h + r\cos t \\ y &= k + r\sin t \end{aligned} \) \( t \in [0, 2\pi) \) |
| Ellipse | \( \dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1 \) | \( \begin{aligned} x &= h + a\cos t \\ y &= k + b\sin t \end{aligned} \) \( t \in [0, 2\pi) \) |
| Hyperbola | \( \dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1 \) | \( \begin{aligned} x &= h + a\sec t \\ y &= k + b\tan t \end{aligned} \) \( t \in \left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right) \cup \left(\tfrac{\pi}{2}, \tfrac{3\pi}{2}\right) \) |
A vector-valued function is a function $\vec{r} : \mathbb{R} \to \mathbb{R}^3$. It eats real numbers and spits out vectors: \[\vec{r}(t) = \vc{x(t), y(t), z(t)} = x(t)\iv + y(t)\jv + z(t)\kv\]
The component functions $x(t), y(t), z(t)$ are parametrics in disguise. The space curve traced by $\vec{r}$ is the path swept out by the tip of the position vector $\vec{r}(t)$.
Think Input a time $t$, output a position.
The domain of $\vec{r}(t)$ is the set of all $t$ you can plug in and get a vector out. That is, it is the intersection of the domains of the component functions.
To parametrize the curve where two surfaces meet:
If $\vec{r}(t) = \vc{f(t), g(t), h(t)}$, then \[\lim_{t\to a}\vec{r}(t) = \vc{\lim_{t\to a} f(t),\ \lim_{t\to a} g(t),\ \lim_{t\to a} h(t)}\] provided the limits of the component functions exist.
Let $C$ be the curve traced by $\vec{r}(t)$. The derivative $\vec{r}\,'(t)$ is defined \[\vec{r}\,'(t) = \dfrac{d\vec{r}}{dt} = \lim_{h\to 0}\dfrac{\vec{r}(t + h) - \vec{r}(t)}{h}\] if this limit exists. $\vec{r}\,'(t)$ is called the tangent vector to $C$.
The tangent line to $C$ at $\vec{r}(t_0)$ is the line parallel to $\vec{r}\,'(t_0)$ at the point of tangency.
Think If $\vec{r}(t)$ is the track of a rollercoaster and you are sitting at the tip of $\vec{r}(t)$, then the direction of $\vec{r}\,'(t)$ is your line of sight.
If $\vec{r}(t) = \vc{f(t), g(t), h(t)} = f(t)\iv + g(t)\jv + h(t)\kv$ where $f, g$ and $h$ are differentiable functions, then \[\vec{r}\,'(t) = \vc{f'(t), g'(t), h'(t)} = f'(t)\iv + g'(t)\jv + h'(t)\kv\]
A unit vector that has the same direction as $\vec{r}\,'(t)$ is \[\vec{T}(t) = \dfrac{\vec{r}\,'(t)}{|\vec{r}\,'(t)|}\]
If $\vec{r}(t)$ describes the position vector of an object, then \[\vec{v}(t) = \vec{r}\,'(t)\] describes the rate of change of position of the object with respect to time.
The speed of the object is the magnitude $|\vec{v}(t)| = |\vec{r}\,'(t)|$. Speed is a scalar function; velocity is a vector function.
The acceleration of the object is \[\vec{a}(t) = \vec{v}\,'(t) = \vec{r}\,''(t)\]
$\vec{a}(t)$ records how the tip of $\vec{v}(t)$ is changing: its direction says which way the tip of $\vec{v}(t)$ is being dragged next, and its magnitude says how fast the tip of $\vec{v}(t)$ is changing.
Acceleration changes velocity in two different ways:
Suppose $\vec{u}$ and $\vec{v}$ are differentiable vector functions, $c \in \mathbb{R}$ and $f$ is a real-valued function.
Let $\vec{r}(t) = \vc{x(t), y(t), z(t)}$ be a continuous vector function. Then \[\int_a^b \vec{r}(t)\,dt = \vc{\int_a^b x(t)\,dt,\ \int_a^b y(t)\,dt,\ \int_a^b z(t)\,dt}\]
The indefinite integral is \[\int \vec{r}(t)\,dt = \vc{\int x(t)\,dt,\ \int y(t)\,dt,\ \int z(t)\,dt} + \vec{c}\] where $\vec{c} = \vc{c_1, c_2, c_3}$ is a constant vector.
Think Just like differentiation, integrate each component separately.
If a curve $C$ has vector equation $\vec{r}(t) = \vc{x(t), y(t), z(t)}$ where $a \leq t \leq b$ and the curve is traversed exactly once as $t$ increases from $a$ to $b$, then the arc length of $C$ is \[L = \int_a^b \sqrt{\left[x'(t)\right]^2 + \left[y'(t)\right]^2 + \left[z'(t)\right]^2}\,dt = \int_a^b \sqrt{\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2 + \left(\dfrac{dz}{dt}\right)^2}\,dt\]
Since $\vec{r}\,'(t) = \vc{x'(t), y'(t), z'(t)}$, we can simply write \[L = \int_a^b \left|\vec{r}\,'(t)\right|\,dt\]
In the staircase paradox, a staircase approximation to the diagonal of the unit square has length 2 at every stage, yet the diagonal has length $\sqrt{2}$.
The issue is that tangency to the curve is not preserved: the staircase's corners are not on the line you are trying to estimate. In a proper arc length calculation, all estimating points are on the curve, so tangency is preserved under the limit.
Suppose the curve $C$ is traversed exactly once by $\vec{r}(t) = \vc{x(t), y(t), z(t)}$ where $t \in [a,b]$ as $t$ increases from $a$ to $b$. The arc length function is \[s(t) = \int_a^t \left|\vec{r}\,'(u)\right|\,du\]
By the Fundamental Theorem of Calculus, \[\dfrac{ds}{dt} = \left|\vec{r}\,'(t)\right|\] which says "the rate of change of arc length with respect to $t$ is exactly the speed of $\vec{r}(t)$."