Q216 / 270 · NDA 2023 (Phase 1) · Mathematics

Paragraph: Let $\vec{a}=3 \hat{i}+3 \hat{j}+3 \hat{k}$ and $\vec{c}=\hat{j}-\hat{k}$. Let $\vec{b}$ be such that $\vec{a} \cdot \vec{b}=27$ and $\vec{a} \times \vec{b}=\overrightarrow{9 c}$
Question: What is the angle between $(\vec{a}+\vec{b})$ and $\vec{c}$ ?

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

Solution

$\vec{a}+\vec{b}=8 \hat{i}+5 \hat{j}+5 \hat{k}$ $\vec{c}=\hat{j}-\hat{k}$ $\Rightarrow \quad(\vec{a}+\vec{b}) \cdot \vec{c}=|\vec{a}+\vec{b}| \cdot|\vec{c}| \cdot \cos \theta$ $\Rightarrow 5-5=|\vec{a}+\vec{b}| \cdot|\vec{c}| \cos \theta$ $\Rightarrow \quad \cos \theta=0$ $\therefore \quad \theta=\frac{\pi}{2}$

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

Relevant Questions