1. Notation: Vectors are often written as $\mathbf{a}$, or in component form, $\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}$ in 2D or $\\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ in 3D.
2. Magnitude: The length of a vector $\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}$ is given by:

3. Direction: Defined by the angle the vector makes with a reference axis.
4. Addition/Subtraction: Vectors can be added or subtracted component-wise:

5. Scalar Multiplication: A vector can be multiplied by a scalar, which changes its magnitude but not its direction.

A vector is a set of directions on how to get from one place to another. For example, the vector $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ means move $2$ in the $x (or i)$ direction, then $3$ in the $y (or j)$ direction.

There are two types of vectors commonly used: displacement vectors and position vectors.

• A position vector tells you how to get from the origin $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$ to a point.
• A displacement vector tells you how to get between any two points. Both types look and should be treated identically.

Example Problem:

Find the vector that takes you from $A\begin{pmatrix} 2 \\ 7 \end{pmatrix}$ to $B\begin{pmatrix} 6 \\ -2 \end{pmatrix}$.

6. The vector $\overrightarrow{AB}$ means "the vector taking you from $A$ to $B$."
7. To find $\overrightarrow{AB}$:

• Use the formula: $\overrightarrow{AB} = B - A$
• $\overrightarrow{AB} = \begin{pmatrix} 6 \\ -2 \end{pmatrix} - \begin{pmatrix} 2 \\ 7 \end{pmatrix}$
• $\overrightarrow{AB} = \begin{pmatrix} 4 \\ -9 \end{pmatrix}$

Example:

This is a unit vector because:

Example Problem: Find a unit vector parallel to $\begin{pmatrix} 7 \\ 3 \end{pmatrix}$ Note: The answer will be a multiple of $\begin{pmatrix} 7 \\ 3 \end{pmatrix}$.

8. Find the magnitude of the original vector:

9. Shorten the vector by this scale factor to get the unit vector:

Note: $a \left(\begin{array}{c} x \\ y \end{array}\right)$ means $\left(\begin{array}{c} ax \\ ay \end{array}\right)$.