~~NOTOC~~
====== Vector ======
A **vector** (specifically a **Euclidean vector**) is a geometric object with direction and magnitude. A vector is often represented as a directed line segment defined by an initial point $A$ with a terminal point $B$. 

Vector notation varies; a typical vector representation is $\vec{AB}$, symbolizing the vector from point $A$ to point $B$ (i.e. what is needed to "carry" point $A$ to point $B$). A vector may also be represented by a single symbol, e.g. $\vec{u}$.

===== Properties =====

==== Addition/subtraction ====
Adding/subtracting vectors involves the addition/subtraction of each vector coordinate. As long as both vectors have matching dimensions, any two vectors can be added/subtracted to compute a resultant vector.

$\vec{x} = \begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix},$
$\vec{y} = \begin{bmatrix}y_1 \\ y_2 \\ y_3 \end{bmatrix}$
$\quad\vec{x} + \vec{y} = \begin{bmatrix}x_1 + y_1 \\ x_2 + y_2 \\ x_3 + y_3 \end{bmatrix}$
$\quad\vec{x} - \vec{y} = \begin{bmatrix}x_1 - y_1 \\ x_2 - y_2 \\ x_3 - y_3 \end{bmatrix}$

\\
\\
<color #00a2e8>**(Ex. 1)**</color> Solve $\vec{x} + \vec{y}$
\\
\\
$\vec{x} = \begin{bmatrix}5 \\ -3 \\ 7 \end{bmatrix},$
$\vec{y} = \begin{bmatrix}0 \\ 15 \\ -2 \end{bmatrix}$
<details><summary><color #00a2e8>**[Show Answer]**</color></summary> $\vec{x} + \vec{y} = \begin{bmatrix}5 + 0 \\ -3 + 15 \\ 7 + -2 \end{bmatrix} = \begin{bmatrix}5 \\ 12 \\ 5 \end{bmatrix}$ </details>
\\
\\
<color #00a2e8>**(Ex. 2)**</color> Solve $\vec{x} - \vec{y}$
\\
\\
$\vec{x} = \begin{bmatrix}-3 \\ 13 \\ 6 \end{bmatrix},$
$\vec{y} = \begin{bmatrix}-9 \\ -4 \\ 12 \end{bmatrix}$
<details><summary><color #00a2e8>**[Show Answer]**</color></summary> $\vec{x} - \vec{y} = \begin{bmatrix}-3 - -9 \\ 13 - -4 \\ 6 - 12 \end{bmatrix} = \begin{bmatrix}6 \\ 17 \\ -6 \end{bmatrix}$ </details>
==== Scaling ====
To scale a vector is to change magnitude without changing direction. A vector is scaled when multiplied by a positive real coefficient.

$\displaystyle\{k \in \mathbb{R} \mkern9mu | \mkern9mu 0 < k < \infty\},$
$\quad\vec{x} = \begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix},$
$\quad k\vec{x} = \begin{bmatrix}kx_1 \\ kx_2 \\ kx_3 \end{bmatrix}$

\\
\\
<color #00a2e8>**(Ex. 3)**</color> Find a real value $k$ such that  $||k * \vec{x}|| = 2.5$
\\
\\
$\vec{x} = \begin{bmatrix}3 \\ -4 \\ 0 \end{bmatrix}$
<details><summary><color #00a2e8>**[Show Answer]**</color></summary> $\displaystyle || \vec{x} || = \sqrt{3^2 + (-4)^2} = 5$\\
$\displaystyle 2.5/5 = 1/2, \quad \displaystyle k || \vec{x} || = \frac{1}{2} || \vec{x} || = 2.5$\\
$k = 1/2$ </details>
\\
\\
==== Normalization ====

==== Dot product ====

==== Cross product ====



==== See also ====
  * [[article:eigen|Eigenvector]]
  * [[article:linear_algebra|Linear algebra]]