Vector

definition

A vector is an array of real numbers $\vec{x} = [x_{1}, x_{2}, \dots, x_{n}]$ , where $x_{1}, x_{2}, \dots, x_{n} \in R$

definition

$R^{n} = {[\begin{matrix} x_{1} \\ x_{2} \\ ⋮ \\ x_{n} \end{matrix}] : x_{1}, \dots, x_{n} \in R}$

Notice the : instead of |

Direction and magnitude/length/norm
Projections

\vec{x} = [\begin{matrix} x_{1} \\ ⋮ \\ x_{n} \end{matrix}], \vec{y} = [\begin{matrix} y_{1} \\ ⋮ \\ y_{n} \end{matrix}] \in R^{n}

operations on differently shaped vectors are undefined
$R^{3}$ x1 is towards you, x2 is to right, x3 is up
$\vec{e_{i}}$ is a "standard" basis vector with 0s and a 1 in position i
normalization: $\vec{x} \to \hat{x} = \frac{1}{‖ \vec{x} ‖}$
Vectors are orthogonal if their dot product is 0. They are perpendicular if they make a $90 °$ angle. The difference is with the $\vec{0}$

Vector Operations

Dot Product
Cross Product
Addition
- commutative
- associative
Scaling
- Associative

Magnitude

aka norm or distance or length
norm of $\vec{x}$ is $| | \vec{x} | | = \sqrt{x_{1}^{2} + x_{2}^{2} + \dots + x_{n}^{2}}$
$‖ \vec{x} ‖ \geq 0$
- $‖ \vec{x} ‖ = 0$ iff $\vec{x} = \vec{0}$
$‖ c \vec{x} ‖ = | c | ‖ \vec{x} ‖$

proof

$\begin{aligned} ‖ c \vec{x} ‖ & = ‖ c [\begin{array}{c} x_{1} \\ ⋮ \\ x_{n} \end{array}] ‖ \\ = ‖ [\begin{array}{c} c x_{1} \\ ⋮ \\ c x_{n} \end{array}] ‖ \\ = \sqrt{(c x_{1})^{2} + \dots + (c x_{n})^{2}} \\ = \sqrt{c^{2} (x_{1}^{2} + x_{n}^{2})} \\ = | c | \sqrt{x_{1}^{2} + \dots + x_{n}^{2}} \\ = | c | ‖ \vec{x} ‖, ◼ \end{aligned}$

$‖ \vec{x} + \vec{y} ‖ \leq ‖ \vec{x} ‖ + ‖ \vec{y} ‖$ (triangle inequality)

\Vert \vec{x}-\vec{y} \Vert & =\Vert -1(\vec{y}-\vec{x}) \Vert \
& =|-1|\Vert \vec{y}-\vec{x} \Vert \
& =\Vert \vec{y}-\vec{x} \Vert
\end

## Position Vector - Tail is at origin - Aka algebraic vector - If a vector is just a variable, it's a position vector - Can be expressed as $(a, b, c)$ - {component form} or $a\hat{i}+b\hat{j}+\hat{c}k$ - {unit vector form}{#1705549184819} - $|\vec{u}|=\sqrt{ a^{2}+b^{2}+c^{2} }$ > [!NOTE]- Proof >

\begin{align}
|\vec{u}| & =\sqrt{ |a+b|^{2}+c^{2} } \
|a+b| & =\sqrt{ a^{2}+b^{2} } \
|\vec{u}| & =\sqrt{ (\sqrt{ a^{2}+b^{2} })^{2}+c^{2} } \
|\vec{u}| & =\sqrt{ a^{2}+b^{2}+c^{2} }
\end

- $RAA=\arctan\left( | \frac{b}{a}| \right)$ - Vector $\vec{AB}$ expressed as position vectors is >> $\vec{OB}-\vec{OA}$ {#1705549372900} - Three points are collinear if and only if any two vectors between two points are scalar multiples of each other ## In 3-space - A position vector $\vec{OP}=(a,b,c)$ makes angles $\alpha,\beta,\gamma$ with the x, y, and z axes, respectively - $0\textdegree\leq\alpha\leq 180\textdegree, 0\textdegree\leq\beta\leq 180\textdegree, 0\textdegree\leq\gamma\leq 180\textdegree$ - $|\vec{OP}|=\sqrt{ a^{2}+b^{2}+c^{2} }$ - $\alpha=\cos ^{-1}\left( \frac{a}{\sqrt{ a^{2}+b^{2}+c^{2} }} \right),\beta=\cos ^{-1}\left( \frac{b}{\sqrt{ a^{2}+b^{2}+c^{2} }} \right),\gamma=\cos ^{-1}\left( \frac{c}{\sqrt{ a^{2}+b^{2}+c^{2} }} \right)$ - Components of $\hat{u}$ in terms of $\alpha, \beta, \gamma$ -

\begin{align}
\hat{u} & =\frac{\vec{OP}}{|\vec{OP}|} \
& =\frac{1}{\sqrt{ a^{2}+b^{2}+c^{2} }} (a,b,c) \
& =\left( \frac{a}{\sqrt{ a^{2}+b^{2}+c^{2} }}, \frac{b}{\sqrt{ a^{2}+b^{2}+c^{2} }}, \frac{c}{\sqrt{ a^{2}+b^{2}+c^{2} }} \right) \
& =(\cos\alpha, \cos\beta, \cos\gamma)
\end

- P r o v e $ \cos^{2} α + \cos^{2} β + \cos^{2} γ = 1 $ -

\begin{align}
\hat{u} & =(\cos\alpha, \cos\beta, \cos\gamma) \
|\hat{u}| & =\sqrt{ \cos ^{2}\alpha + \cos ^{2}\beta + \cos ^{2}\gamma } \
1 & =\sqrt{ \cos ^{2}\alpha + \cos ^{2}\beta + \cos ^{2}\gamma } \
1 & =\cos ^{2}\alpha + \cos ^{2}\beta + \cos ^{2}\gamma
\end