Vocabulary and Review Problems

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Vocabulary and Review Problems

Algebraic properties of vectors

The following properties hold for vectors $\vec {u}$ , $\vec {v}$ and $\bf w$ in $\RR ^n$ and scalars $k$ and $p$ in $\RR$ .

(a): Commutative Property of Addition

$\vec {u}+\vec {v}=\vec {v}+\vec {u}$
(b): Associative Property of Addition

$(\vec {u}+\vec {v})+{\bf w}=\vec {u}+(\vec {v}+{\bf w})$
(c): Existence of Additive Identity:

There exists a vector $\vec {0}$ such that $\vec {u}+\vec {0}=\vec {u}$
(d): Existence of Additive Inverse:

For every vector $\vec {u}$ , there exists a vector $-\vec {u}$ such that $\vec {u}+(-\vec {u})=\vec {0}$
(e): Distributive Property over Vector Addition

$k(\vec {u}+\vec {v})=k\vec {u}+k\vec {v}$
(f): Distributive Property over Scalar Addition

$(k+p)\vec {u}=k\vec {u}+p\vec {u}$
(g): Associative Property for Scalar Multiplication

$k(p\vec {u})=(kp)\vec {u}$
(h): Multiplication by 1

$1\vec {u}=\vec {u}$

Angle between vectors

Let $\vec {u}$ and $\vec {v}$ be vectors in $\RR ^n$ , and let $\theta$ be the included angle. Then $\vec {u}\dotp \vec {v}=\norm {\vec {u}}\norm {\vec {v}}\cos \theta$

Cross product

Let $\vec {u=\begin {bmatrix}u_1\\u_2\\u_3\end {bmatrix}}$ and $\vec {v}=\begin {bmatrix}v_1\\v_2\\v_3\end {bmatrix}$ be vectors in $\RR ^3$ . The cross product of $\vec {u}$ and $\vec {v}$ , denoted by $\vec {u}\times \vec {v}$ , is given by $\vec {u}\times \vec {v}=(u_2v_3-u_3v_2)\vec {i}-(u_1v_3-u_3v_1)\vec {j}+(u_1v_2-u_2v_1)\vec {k}=\begin {bmatrix}u_2v_3-u_3v_2\\-u_1v_3+u_3v_1\\u_1v_2-u_2v_1\end {bmatrix}$

Direction vector for a line

Distance between two points

Let $A(a_1, a_2,\ldots ,a_n)$ and $B(b_1, b_2,\ldots ,b_n)$ be points in $\RR ^n$ . The distance between $A$ and $B$ is given by $AB=\sqrt {(a_1-b_1)^2+(a_2-b_2)^2+\ldots +(a_n-b_n)^2}$

Dot product

Let $\vec {u}$ and $\vec {v}$ be vectors in $\RR ^n$ . The dot product of $\vec {u}$ and $\vec {v}$ , denoted by $\vec {u}\dotp \vec {v}$ , is given by $\vec {u}\dotp \vec {v}=\begin {bmatrix}u_1\\u_2\\\vdots \\u_n\end {bmatrix}\dotp \begin {bmatrix}v_1\\v_2\\\vdots \\v_n\end {bmatrix}=u_1v_1+u_2v_2+\ldots +u_nv_n$

Dot product properties

The following properties hold for vectors $\vec {u}$ , $\vec {v}$ and $\vec {w}$ in $\RR ^n$ and scalar $k$ in $\RR$ .

(a): $\vec {u}\dotp \vec {v}=\vec {v}\dotp \vec {u}$
(b): $(\vec {u}+\vec {v})\dotp \vec {w}=\vec {u}\dotp \vec {w}+\vec {v}\dotp \vec {w}$
(c): $\vec {u}\dotp (\vec {v}+\vec {w})=\vec {u}\dotp \vec {v}+\vec {u}\dotp \vec {w}$
(d): $(k\vec {u})\dotp \vec {v}=k(\vec {u}\dotp \vec {v})=\vec {u}\dotp (k\vec {v})$
(e): $\vec {u}\dotp \vec {u}\geq 0$ , and $\vec {u}\dotp \vec {u}=0$ if and only if $\vec {u}={\bf 0}$ .
(f): $\norm {\vec {u}}^2=\vec {u}\dotp \vec {u}$

Equation of a plane

The plane through $P_{0}(x_{0}, y_{0}, z_{0})$ with a normal vector $\vec {n} = \begin {bmatrix} a\\ b\\ c \end {bmatrix}\neq \vec {0}$ is given by $a(x - x_{0}) + b(y - y_{0}) + c(z - z_{0}) = 0$

Head-to-tail rule for vector addition

Length (norm, magnitude) of a vector

Let $\vec {v}=\begin {bmatrix}v_1\\ v_2\\ \vdots \\v_n\end {bmatrix}$ be a vector in $\RR ^n$ , then the length, or the magnitude, of $\vec {v}$ is given by $\norm {\vec {v}}=\sqrt {v_1^2+v_2^2+\ldots +v_n^2}$

Normal vector

Orthogonal vectors

Let $\vec {u}$ and $\vec {v}$ be vectors in $\RR ^n$ . We say $\vec {u}$ and $\vec {v}$ are orthogonal if $\vec {u}\dotp \vec {v}=0$ .

Parallel vectors

Parallelogram rule for vector addition

Parametric equations of a line

Let $\vec {v}=\begin {bmatrix}v_1\\v_2\\\vdots \\v_n\end {bmatrix}$ be a direction vector for line $l$ in $\RR ^n$ , and let $(a_1, a_2,\ldots , a_n)$ be an arbitrary point on $l$ . Then the following parametric equations describe $l$ : $x_1=v_1t+a_1$ $x_2=v_2t+a_2$ $\vdots$ $x_n=v_nt+a_n$

Projection of a vector onto a vector

Let $\vec {v}$ be a vector, and let $\vec {d}$ be a non-zero vector. The projection of $\vec {v}$ onto $\vec {d}$ is given by $\mbox {proj}_{\vec {d}}\vec {v}=\left (\frac {\vec {v}\dotp \vec {d}}{\norm {\vec {d}}^2}\right )\vec {d}$

Pythagoras’ Theorem

Scalar multiplication

Let $\vec {u}=\begin {bmatrix} u_1\\ u_2\\ \vdots \\ u_n \end {bmatrix}$ be a vector in $\RR ^n$ , and let $k$ be a scalar, then $k\vec {u}=k\begin {bmatrix} u_1\\ u_2\\ \vdots \\ u_n \end {bmatrix}=\begin {bmatrix} ku_1\\ ku_2\\ \vdots \\ ku_n \end {bmatrix}$

Standard unit vectors

Let $\vec {e}_i$ denote a vector that has $1$ as the $i^{th}$ component and zeros elsewhere. In other words, $\vec {e}_i=\begin {bmatrix} 0\\ 0\\ \vdots \\ 1\\ \vdots \\ 0 \end {bmatrix}$ where $1$ is in the $i^{th}$ position. We say that $\vec {e}_i$ is a standard unit vector of $\RR ^n$ .

Unit vector

Vector

Vector addition

Let $\vec {u}=\begin {bmatrix} u_1\\ u_2\\ \vdots \\ u_n \end {bmatrix}$ and $\vec {v}=\begin {bmatrix} v_1\\ v_2\\ \vdots \\ v_n \end {bmatrix}$ be vectors in $\RR ^n$ . We define $\vec {u}+\vec {v}$ by $\vec {u}+\vec {v}=\begin {bmatrix} u_1\\ u_2\\ \vdots \\ u_n \end {bmatrix}+\begin {bmatrix} v_1\\ v_2\\ \vdots \\ v_n \end {bmatrix}=\begin {bmatrix} u_1+v_1\\ u_2+v_2\\ \vdots \\ u_n+v_n \end {bmatrix}$

Zero vector

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norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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