Dot Product

Physicist's Dot Product: (dot product gives the angle) for \overrightarrow{u} and \overrightarrow{v} with 0 \leq \theta \leq \pi as the angle, then \overrightarrow{u} \cdot \overrightarrow{v} = \|\overrightarrow{u}\| \|\overrightarrow{v}\| \cos{\theta}

Property of Dot Product

  1. u \cdot v = v \cdot u
  2. u \cdot (v + w) = u \cdot v + u\cdot w
  3. C(u \cdot v) = (Cu) \cdot v = u \cdot (Cv)
  4. v \cdot v = \|v\|^2

Law of Cosines: C^2 = A^2 + B^2 - 2AB\cos{c} Scalar Projection: |comp_{\overrightarrow{u}}(v)| = \|proj_{\overrightarrow{u}}(v)\| = \frac{|\overrightarrow{u}\cdot \overrightarrow{v}|}{\|\overrightarrow{u}\|} Projection of Vectors: projection of \overrightarrow{v} onto \overrightarrow{u} is proj_{\overrightarrow{u}}(v) = (\cos{\theta} \cdot \|v\|) \cdot \frac{\overrightarrow{u}}{\|u\|} = \frac{\overrightarrow{u}\cdot \overrightarrow{v}}{\|\overrightarrow{u}\|^2}\cdot \overrightarrow{u}

Cross Product

Cross Product (\times): perpendicular (\perp) to all input vectors

\overrightarrow{u} \times \overrightarrow{v} v_i v_j v_k
u_i 0 k -j
u_j -k 0 i
u_k j -i 0

Formal Determinant: \det \begin{pmatrix} \overrightarrow{i} \overrightarrow{j} \overrightarrow{k}\\ u_1 u_2 u_3\\ v_1 v_2 v_3\\ \end{pmatrix} = \overrightarrow{i} \det \begin{pmatrix} \overrightarrow{u_2} \overrightarrow{u_3}\\ \overrightarrow{v_2} \overrightarrow{v_3}\\ \end{pmatrix} - \overrightarrow{j} \det \begin{pmatrix} \overrightarrow{u_1} \overrightarrow{u_3}\\ \overrightarrow{v_1} \overrightarrow{v_3}\\ \end{pmatrix} + \overrightarrow{k} \det \begin{pmatrix} \overrightarrow{u_2} \overrightarrow{u_2}\\ \overrightarrow{v_2} \overrightarrow{v_2}\\ \end{pmatrix}

\begin{bmatrix} u_1\\ u_2\\ u_3\\ \end{bmatrix} \times \begin{bmatrix} v_1\\ v_2\\ v_3\\ \end{bmatrix} = \det(\begin{bmatrix} \overrightarrow{i} & u_1 & v_1\\ \overrightarrow{j} & u_2 & v_2\\ \overrightarrow{k} & u_3 & v_3\\ \end{bmatrix}) = \overrightarrow{i}(u_2v_3 - u_3v_2) - \overrightarrow{j}(u_1v_3 - u_3v_1) + \overrightarrow{k}(u_1v_2 - u_2v_1)

Determinant: for 2 by 2, \det \begin{pmatrix} a b\\ c d\\ \end{pmatrix} = ad - bc

Properties of Cross Product:

Angle: \sin(\theta) = \frac{\|\overrightarrow{u} \times \overrightarrow{v}\|}{\|\overrightarrow{u}\|\|\overrightarrow{v}\|}

Area of Parallelgram: \|\overrightarrow{u} \times \overrightarrow{v}\| (\overrightarrow{u} \times \overrightarrow{v} is positive if \overrightarrow{u} rotate in positive dirrection to get \overrightarrow{v}, \overrightarrow{i} \times \overrightarrow{j} is positive. Since cross product is determinant is area.)

Torque: \tau = \overrightarrow{r} \times \overrightarrow{F} where \overrightarrow{r} is the radius and \overrightarrow{F} is the force.

Line & Plane

Line: a point p = (x_0, y_0, z_0) and a dirrection vector \langle{a, b, c}\rangle, then a line is \begin{cases} x = x_0 + at\\ y = y_0 + bt\\ z = z_0 + ct\\ \end{cases} or \langle{x, y, z}\rangle = \langle{x_0, y_0, z_0}\rangle + t\langle{a, b, c}\rangle

Plane: \pm \overrightarrow{n} \cdot \overrightarrow{PQ} = 0

Angle between Two Plane: Angle (0 < \theta < \frac{\pi}{2}) between P_1 and P_2 with normal vectors n_1 and n_2 is \cos{\theta} = \frac{|\overrightarrow{n_1} \cdot \overrightarrow{n_2}|}{\|n_1\|\|n_2\|}

Distance between a Point and Line & Plane

Distance Between Line (Q, \overrightarrow{v}) and point P is: \frac{\|\overrightarrow{QP} \times \overrightarrow{v}\|}{\|\overrightarrow{v}\|}

Distance between Plane (Q, \overrightarrow{n}) point P is: d = \|\text{proj}_n{\overrightarrow{QP}}\| = |\text{comp}_n \overrightarrow{QP}| = \frac{|\overrightarrow{QP} \cdot n|}{\|n\|}

Conic Section





Vector Valued Function

Limit: limit exists if both side limit exists Continuous: limit exists and equal function value

Dot Product of Vector Valued Function: \overrightarrow{r_1} \cdot \overrightarrow{r_2} = f_1f_2 + g_1g_2 + h_1h_2

Cross Product Derivative: (\overrightarrow{r_1}(t) \times \overrightarrow{r_2}(t))' = \overrightarrow{r_1}(t) \times \overrightarrow{r_2}'(t) + \overrightarrow{r_1}'(t) \times \overrightarrow{r_2}(t)

Arc Length

In \mathbb{R}^2, \text{Arc Length} = \int_{x=a}^{x=b} \sqrt{(f'(x)^2 + 1)} dx In \mathbb{R}^3, \begin{align} \text{Arc Length} &= \int_{t=a}^{t=b} \sqrt{(f'(t))^2 + (g'(t))^2 + (h'(t))^2} dt\\ &= \int_{t=a}^{t=b} \|\overrightarrow{r}'(t)\| dt\\ \end{align}

Arc Length Function: function computes arc length from t=a to t=t given \overrightarrow{r}(t) is \S(t) = \int_a^t \|\overrightarrow{r}'(u)\| du.

\overrightarrow{T}(t) = \frac{\overrightarrow{r}'(t)}{\|\overrightarrow{r}'(t)\|} = \frac{\overrightarrow{r}'(t)}{\|\frac{dS}{dt}\|}

Curvature: \kappa(S) = \|\frac{d \overrightarrow{T}}{dS}\| = \|\overrightarrow{T'}(S)\| for \overrightarrow{r}(S) (parameterized by arc length S)

\overrightarrow{v} \cdot \overrightarrow{w} = 0 \implies \|\overrightarrow{v} \times \overrightarrow{w}\| = \|\overrightarrow{v}\|\|\overrightarrow{w}\|

Principal Unite Normal Vector: \overrightarrow{N}(t) = \frac{\overrightarrow{T}'(t)}{\|\overrightarrow{T}'(t)\|}

Binormal Vector: \overrightarrow{B}(t) = \overrightarrow{T}(t) \times \overrightarrow{N}(t)

Acceleration Components on Curve:

|F_{\text{centripital}}| = \frac{mv^2}{r}


Level Curve: level curve at z=c is the curve in R^2 of z = f(x, y). Contor Map: several labeled level curves on same coordinate axes (ie. multiple level curves in the same dirrection) Vertical Trace: making one independent (instead of dependent) variable constant

If z = f(x, y), the limit of f(x, y) as (x, y) approaches (a, b) is C, denoted \lim_{(x, y) \rightarrow (a, b)} f(x, y) = C provided that for all \epsilon so that when 0 < \text{dist}((x, y), (a, b)) < S, |f(x, y) - C| < S

Limits and Continuity

Limit in 3D: if 0 \leq \text{distance}((x, y), (a, b)) < \delta \implies 0 \leq |f(x, y)-L| < \epsilon the limit of z = f(x, y) is \lim_{(x, y) \rightarrow (a, b)} f(x, y) = L

Find the Limit when Exists

  1. try to plug in
  2. try to simplify, cancel denominators, plug in
  3. If get \frac{0}{0}, show \lim_{(x, y) \rightarrow (a, b)} f(x, y) does not exists
  4. \frac{\text{something}}{0} means derivative DNE

Prove Limit DNE:

  1. Write arbitrary x, y, z, ... all in terms of polynomials of one variable, say x.
  2. Find some polynomials X_1, X_2, Y_1, Y_2 such that \lim_{(X_1, Y_2, ...) \rightarrow (a, b, ...)} f(x, y, ...) \downarrow \neq \lim_{(X_2, Y_2, ...) \rightarrow (a, b, ...)} f(x, y, ...) \downarrow and none of them are \frac{0}{0} \uparrow

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