Q96 / 150 · MHT CET 2022 (07 Aug Shift 1) · Mathematics

The vector projection of $\bar{b}$ on, $\bar{a}=3 \hat{i}+2 \hat{j}+5 \hat{k}$ and $\bar{b}=7 \hat{i}-5 \hat{j}-\hat{k}$

  • A
    $\frac{6(3 \hat{i}+2 \hat{j}+5 \hat{k})}{\sqrt{38}}$
  • B
    $\frac{3(3 \hat{i}+2 \hat{j}+5 \hat{k})}{38}$
  • Correct answer
    C
    $\frac{3(3 \hat{i}+2 \hat{j}+5 \hat{k})}{19}$
  • D
    $\frac{3(3 \hat{i}+2 \hat{j}+5 \hat{k})}{\sqrt{38}}$

Solution

The vector projection of $\vec{b}$ on $\vec{a}$ $\begin{aligned} & =\left(\frac{\vec{b} \cdot \vec{a}}{|\vec{a}|}\right) \hat{a} \\ & =\left(\frac{\vec{b} \cdot \vec{a}}{|\vec{a}|^2}\right) \vec{a} \\ & =\left(\frac{7 \times 3+(-5) \times 2+(-1) \times 5}{3^2+2^2+5^2}\right)(3 \hat{i}+2 \hat{j}+5 \hat{k}) \\ & =\frac{6}{38}(3 \hat{i}+2 \hat{j}+5 \hat{k}) \\ & =\frac{3}{19}(3 \hat{i}+2 \hat{j}+5 \hat{k})\end{aligned}$

Hence, the correct answer is option: 'C'.

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