# Example: Vector subtraction

## Understanding the situation

Subtraction of vectors works similar to addition, but to subtract a vector, $\overrightarrow{b}$ , you add the vector $-\overrightarrow{b}$; like adding $\overrightarrow{b}$ with all the signs of its components reversed. Here's an example.

## Presenting a sample problem

Given that

$\overrightarrow{a}=\hat{i}+2\hat{j}$  and  $\overrightarrow{b} = -3\hat{j}$

find the vector

$$\overrightarrow{c} = 2\overrightarrow{a} - \overrightarrow{b}$$

using both algebraic and geometric methods.

## Solving this problem

#### Solving it algebraically

To solve it algebraically, we just replace the vectors by how they are expressed in terms of the unit vectors and then rearrange to collect terms.

$$2\overrightarrow{a} - \overrightarrow{b} = 2\times(1\hat{i} + 2\hat{j}) - (-3\hat{j})$$

$$2\overrightarrow{a} - \overrightarrow{b} =(2\times 1 +0)\hat{i} + (2\times 2\ +3)\hat{j}) = 2\hat{i} + 7\hat{j}$$

To subtract $\overrightarrow{b}$ all we had to do was change the -3 to a +3 since a minus times a minus is a plus. The rest is the same as adding.

#### Solving it geometrically

We still want to use a displacement metaphor as we did in adding vectors, but now we have to add $-\overrightarrow{b}$ instead of just $\overrightarrow{b}$. The geometry looks like shown at the right. We've drawn the original vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ in lightly in blue and red respectively. We've then doubled $\overrightarrow{a}$ (deep blue) and flipped $\overrightarrow{b}$ so that it becomes $-\overrightarrow{b}$ and its tail starts at the head of $2\overrightarrow{a}$ -- just as if you had walked from the origin to the head of the vector $2\overrightarrow{a}$ and then started to walk the vector $-\overrightarrow{b}$. The result, shown in green, has 2 units on the x axis and 7 units on the y axis, the same as we got from our algebraic analysis.

Joe Redish 12/26/14

Article 315