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