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. $\vec{u}$.

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.

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

(Ex. 1) Solve $\vec{x} + \vec{y}$

$\vec{x} = \begin{bmatrix}5 \\ -3 \\ 7 \end{bmatrix},$ $\vec{y} = \begin{bmatrix}0 \\ 15 \\ -2 \end{bmatrix}$

(Ex. 2) Solve $\vec{x} - \vec{y}$

$\vec{x} = \begin{bmatrix}-3 \\ 13 \\ 6 \end{bmatrix},$ $\vec{y} = \begin{bmatrix}-9 \\ -4 \\ 12 \end{bmatrix}$

To scale a vector is to change magnitude without changing direction. A vector is scaled when multiplied by a positive real coefficient.

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

(Ex. 3) Find a real value $k$ such that $||k * \vec{x}|| = 2.5$

$\vec{x} = \begin{bmatrix}3 \\ -4 \\ 0 \end{bmatrix}$

• Eigenvector
• Linear algebra