Basic Vectors & Column Vectors

What is a Column Vector?

A column vector describes a movement from one point to another. It is written in the form:

$$\begin{pmatrix} x \\ y \end{pmatrix}$$

The top number $(x)$ is the movement horizontally (right or left)

The bottom number $(y)$ is the movement vertically (up or down)

Example:

$$\begin{pmatrix} 4 \\ -2 \end{pmatrix}$$

Means move 4 right and 2 down.

Adding and Subtracting Vectors

To add vectors, add their top numbers and bottom numbers separately:

$$\begin{pmatrix}3 \\ 5\end{pmatrix} + \begin{pmatrix}2 \\ -1\end{pmatrix} = \begin{pmatrix}3+2 \\ 5+(-1)\end{pmatrix} = \begin{pmatrix}5 \\ 4\end{pmatrix}$$

To subtract vectors, subtract each component:

$$\begin{pmatrix}6 \\ 2\end{pmatrix} - \begin{pmatrix}1 \\ 5\end{pmatrix} = \begin{pmatrix}6-1 \\ 2-5\end{pmatrix} = \begin{pmatrix}5 \\ -3\end{pmatrix}$$

Multiplying Vectors by Scalars

A scalar is just a number. Multiply both parts of the vector by the scalar:

$$3 \times \begin{pmatrix}-2 \\ 4\end{pmatrix} = \begin{pmatrix}3\times(-2) \\ 3\times4\end{pmatrix} = \begin{pmatrix}-6 \\ 12\end{pmatrix}$$

Combining Multiple Vectors

If an expression has more than one vector, do it step by step:

Example:

$$2 \begin{pmatrix}4 \\ 1\end{pmatrix} + 3 \begin{pmatrix}-1 \\ 2\end{pmatrix}$$

Step 1: Multiply each vector:

$$\begin{pmatrix}8 \\ 2\end{pmatrix} + \begin{pmatrix}-3 \\ 6\end{pmatrix}$$

Step 2: Add:

$$\begin{pmatrix}8+(-3) \\ 2+6\end{pmatrix} = \begin{pmatrix}5 \\ 8\end{pmatrix}$$

Worked Example

Let $\mathbf{a} = \begin{pmatrix}p \\ 2\end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix}-3 \\ 1\end{pmatrix}$

Given:

$$2\mathbf{a} + 4\mathbf{b} = \begin{pmatrix}2 \\ 12\end{pmatrix}$$

Step 1: Multiply:

$$2\mathbf{a} = \begin{pmatrix}2p \\ 4\end{pmatrix}, \quad 4\mathbf{b} = \begin{pmatrix}-12 \\ 4\end{pmatrix}$$

Step 2: Add vectors:

$$\begin{pmatrix}2p - 12 \\ 4 + 4\end{pmatrix} = \begin{pmatrix}2p - 12 \\ 8\end{pmatrix}$$

Set equal to the given vector:

$$\begin{pmatrix}2p - 12 \\ 8\end{pmatrix} = \begin{pmatrix}2 \\ 12\end{pmatrix}$$

Match components:

• Top: $2p - 12 = 2 \Rightarrow 2p = 14 \Rightarrow p = 7$
• Bottom: $8 = 12 \to$ This is a contradiction, so something in the setup must be incorrect.