 8.3.2 Subtraction of Vectors

8.3.2 Subtraction of Vectors
The subtraction of the vector $\underset{˜}{b}$ from the vector  $\underset{˜}{a}$ is written as  $\underset{˜}{a}-\underset{˜}{b}$ . This operation can be considered as the addition of the vector $\underset{˜}{a}$ with the negative vector of $\underset{˜}{b}$ . Therefore $\underset{˜}{a}-\underset{˜}{b}=\underset{˜}{a}+\left(-\underset{˜}{b}\right).$

Example 1:

In the diagram above, vector and Q divides PR in the ratio of 2 : 3. Find the following vectors in terms of

Solution:
(a)
$\begin{array}{l}\stackrel{\to }{PR}=\stackrel{\to }{PO}+\stackrel{\to }{OR}\\ \text{}=-\stackrel{\to }{OP}+\stackrel{\to }{OR}\\ \text{}=-\underset{˜}{p}+\underset{˜}{r}\end{array}$

(b)
$\begin{array}{l}\stackrel{\to }{OQ}=\stackrel{\to }{OP}+\stackrel{\to }{PQ}\\ \text{}=\stackrel{\to }{OP}+\frac{2}{5}\stackrel{\to }{PR}\\ \text{}=\underset{˜}{p}+\frac{2}{5}\left(-\underset{˜}{p}+\underset{˜}{r}\right)\\ \text{}=\underset{˜}{p}-\frac{2}{5}\underset{˜}{p}+\frac{2}{5}\underset{˜}{r}\\ \text{}=\frac{3}{5}\underset{˜}{p}+\frac{2}{5}\underset{˜}{r}\end{array}$

(c)
$\begin{array}{l}\stackrel{\to }{QM}=\stackrel{\to }{QO}+\stackrel{\to }{OM}\\ \text{}=-\stackrel{\to }{OQ}+\stackrel{\to }{OM}\\ \text{}=-\stackrel{\to }{OQ}+\frac{1}{2}\stackrel{\to }{OR}\\ \text{}=-\left(\frac{3}{5}\underset{˜}{p}+\frac{2}{5}\underset{˜}{r}\right)+\frac{1}{2}\underset{˜}{r}\\ \text{}=-\frac{3}{5}\underset{˜}{p}-\frac{2}{5}\underset{˜}{r}+\frac{1}{2}\underset{˜}{r}\\ \text{}=-\frac{3}{5}\underset{˜}{p}+\frac{1}{10}\underset{˜}{r}\end{array}$