Vector Addition and Subtraction

Understanding Vectors in a Plane

Vectors are quantities that have both magnitude and direction. In a plane, vectors are often represented as arrows, where the length represents the magnitude and the arrowhead indicates the direction.

For example, a vector can be written as $\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix}$, where $a_1$ and $a_2$ are components along the x-axis and y-axis respectively.

Vector Addition

• To add two vectors $\mathbf{a}$ and $\mathbf{b}$, you add their corresponding components:

$$\mathbf{a} + \mathbf{b} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix}$$

Vector Subtraction

• To subtract vector $\mathbf{b}$ from vector $\mathbf{a}$, subtract their corresponding components:

$$\mathbf{a} - \mathbf{b} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} - \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 - b_1 \\ a_2 - b_2 \end{pmatrix}$$

Worked Example

Example

Subtract the vector $\mathbf{b} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ from $\mathbf{a} = \begin{pmatrix} 5 \\ 7 \end{pmatrix}$.

Solution:

• $$\mathbf{a} - \mathbf{b} = \begin{pmatrix} 5 \\ 7 \end{pmatrix} - \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 5 - 2 \\ 7 - 3 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$$