Q241 / 270 · NDA 2023 (Phase 2) · Mathematics

What is the length of projection of the vector $\hat{i}+2 \hat{j}+3 \hat{k}$ on the vector $2 \hat{i}+3 \hat{j}-2 \hat{k}$ ?

  • A
    $\frac{1}{\sqrt{17}}$
  • Correct answer
    B
    $\frac{2}{\sqrt{17}}$
  • C
    $\frac{3}{\sqrt{17}}$
  • D
    $\frac{2}{\sqrt{14}}$

Solution

Length of projection of the vector $\hat{i}+2 \hat{j}+3 \hat{k}$ on the vector $2 \hat{i}+3 \hat{j}-2 \hat{k}$ is $\frac{(\hat{i}+2 \hat{j}+3 \hat{k}) \cdot(2 \hat{i}+3 \hat{j}-2 \hat{k})}{|2 \hat{i}+3 \hat{j}-2 \hat{k}|}=\frac{2+6-6}{\sqrt{4+9+4}}=\frac{2}{\sqrt{17}}$

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

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