Question

If $\overline{a}=4\overline{i}+6\overline{j}$ and $\overline{b}=3\overline{j}+4\overline{k}$, then the vector form of the component of $\overline{a}$ along $\overline{b}$ is

• A. $\frac{18(3\overline{j}+4\overline{k})}{10\sqrt{3}}$
• B. $\frac{18(3\overline{j}+4\overline{k})}{25}$
• C. $\frac{18(3\overline{j}+4\overline{k})}{\sqrt{13}}$
• D. $\frac{(3\overline{j}+4\overline{k})}{25}$

Answer 1

The correct option is B $\frac{18(3\overline{j}+4\overline{k})}{25}$

$\begin{array}{c}\overline{a}=4\overline{i}+6\overline{j}\overline{b}=3\overline{j}+4\overline{k}\\ \\ \left|\overrightarrow{b}\right|=\sqrt{9+16}=5\\ \\ \overrightarrow{a}\cdot \overrightarrow{b}=4\times 0+6\times 3+0\times 4=18\end{array}$

Component of $\overrightarrow{a}$ along $\overrightarrow{b}=\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{|\overrightarrow{b}|}=\frac{18}{5}$

Component of $\overrightarrow{a}$ along $\overrightarrow{b}=\frac{18}{5}\hat{b}$

$=\frac{18}{5}\left(\frac{\overrightarrow{b}}{|\overrightarrow{b}|}\right)$

$=\frac{18}{5}\left(\frac{3\hat{j}+4\hat{k}}{5}\right)$

$=\frac{18(3\hat{j}+4\hat{k})}{25}$

Hence, option $B$ is the correct option.