notes/education/math/MATH1060 (trig)/Vectors.md

A vector is a mathematical concept that denotes direction and magnitude. They're often notated using an arrow ($\vec{v}$), or with a bold, lowercase letter. (**v**).


Vectors are often denoted as a matrix with two rows: $\begin{bmatrix}1 \\2\end{bmatrix}$

# Magnitude
The magnitude of a vector is $|\vec{v}| = \sqrt{a^2 + b^2}$

# Direction
The direction of a vector is $\theta = \tan^-1(\frac{b}{a})$.

# Addition
To find $\vec{u} + \vec{v}$, we can put one vector on the end of another vector. The resulting vector will share the same tail as the first vector, and the same head as the second vector.

# Scalar Multiplication
A **scalar** is just a real number. Scalar multiplication is multiplying a vector with a real number. This will scale or shrink a vector, but does not change the direction it points at.

We do not multiply two vectors together.
# Unit Vector
A vector with a magnitude of 1 is a **unit vector**.

If $\vec{v}$ is a nonzero vector, the unit vector can be found using the equation $\vec{u} = \dfrac{1}{|\vec{v}|}\vec{v}$ . In other words, to find a unit vector, divide the vector by its magnitude.

# $i$, $j$ Notation
Every 2d vector has a horizontal component and a vertical component. The horizontal unit vector could be written as $i = < 1, 0 >$, and the vertical unit vector could be written as $j = <0, 1 >$. Every vector can be made up using a combination of these standard unit vectors.