Question

Given the vectors $\overline{a}$ and $\overline{b}$ as follows. Find the projections of $\overline{a}$ on $\overline{b}$ and of $\overline{b}$ on $\overline{a}$.

$\overline{a}=\hat{i}+\hat{j}+\hat{k}$; $\overline{b}=\sqrt{3}\hat{i}+3\hat{j}-2\hat{k}$

• A. $\frac{\sqrt{2}+1}{5},\frac{3+\sqrt{3}}{5}.$
• B. $\frac{\sqrt{3}+1}{4},\frac{3+\sqrt{3}}{3}.$
• C. $\frac{\sqrt{5}+2}{4},\frac{4+\sqrt{3}}{4}.$
• D. $\frac{\sqrt{2}+5}{3},\frac{4+\sqrt{3}}{3}.$

Answer 1

The correct option is B $\frac{\sqrt{3}+1}{4},\frac{3+\sqrt{3}}{3}.$

Given $\overline{a}=\hat{i}+\hat{j}+\hat{k}$; $\overline{b}=\sqrt{3}\hat{i}+3\hat{j}-2\hat{k}$
Projection of $\overline{a}$ on $\overline{b}$ is
$\frac{\overline{a}\cdot \overline{b}}{|\overline{b}|}=\frac{\sqrt{3}+1}{4}$
Projection of $\overline{b}$ on $\overline{a}$ is
$\frac{\overline{a}\cdot \overline{b}}{|\overline{a}|}=\frac{\sqrt{3}+1}{\sqrt{3}}=\frac{3+\sqrt{3}}{3}$
Hence, option $B$.