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**).
If $\vec{v}$ is a vector with the initial point $(x_y,\ y_i)$, and a terminal point $(x_t,\ y_t)$, we can express $\vec{v}$ in component form as $\vec{v} = \langle x_t - x_i,\ y_t, -y_i \rangle$
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.
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.
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.
Every 2d vector has a horizontal component and a vertical component. The horizontal unit vector could be written as $i = \langle 1, 0 \rangle$, and the vertical unit vector could be written as $j = \langle 0, 1 \rangle$ Every vector can be made up using a combination of these standard unit vectors.
# Trigonometric Form
Given a vector $\vec{v}$ with a magnitude $|\vec{v}|$ and direction $\theta$: