Question

$Themagnitudeoftheprojectionof2\hat{i}+3\hat{j}+4\hat{k}onthevector\hat{i}+\hat{j}+\hat{k}willbe\underset{–}{}$

• A. $\sqrt{3}$
• B. $\sqrt{\frac{3}{2}}$
• C. $3\sqrt{3}$
• D. None of the above

Answer 1

The correct option is C

$3\sqrt{3}$

$Let\overrightarrow{a}and\overrightarrow{b}bethetwovectorsasshown$

$Theprojectionof\overrightarrow{b}along\overrightarrow{a}willbeOA=\left|\overrightarrow{b}\right|cos\theta =\left|\overrightarrow{b}\right|\left(\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|}\right)(cos\theta =\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|})=\frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{a}\right|}Somagnitudeofprojectionof2\hat{i}+3\hat{j}+4\hat{k}on\hat{i}+\hat{j}+\hat{k}willbe(2\hat{i}+3\hat{j}+4\hat{k}).\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}=\frac{2}{\sqrt{3}}+\frac{3}{\sqrt{3}}+\frac{4}{\sqrt{3}}=\frac{9}{\sqrt{3}}=3\sqrt{3}units$