Q220 / 270 · NDA 2022 (Phase 2) · Mathematics

The position vectors of vertices $A, B$ and $C$ of triangle $A B C$ are respectively $\hat{j}+\hat{k}, 3 \hat{i}+\hat{j}+5 \hat{k}$ and $3 \hat{j}+3 \hat{k}$. What is angle $C$ equal to?

  • A
    $\frac{\pi}{6}$
  • B
    $\frac{\pi}{4}$
  • C
    $\frac{\pi}{3}$
  • Correct answer
    D
    $\frac{\pi}{2}$

Solution

$\begin{aligned} & \overrightarrow{C A}=(3 \hat{j}+3 \hat{k})-(\hat{j}+\hat{k})=2 \hat{j}+2 \hat{k} \\ & \overrightarrow{C B}=(3 \hat{j}+3 \hat{k})-(3 \hat{i}+\hat{j}+5 \hat{k}) \\ & =-3 \hat{i}+2 \hat{j}-2 \hat{k} \\ & \overrightarrow{C A} \cdot \overrightarrow{C B}=0+4-4=0 \\ & \cos C=\frac{\overrightarrow{C A} \cdot \overrightarrow{C B}}{|\overrightarrow{C A}||\overrightarrow{C B}|}=0 \Rightarrow C=\frac{\pi}{2}\end{aligned}$

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

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