Q217 / 270 · NDA 2022 (Phase 2) · Mathematics

Let $\vec{a}$ and $\vec{b}$ are two unit vectors such that $\vec{a}+2 \vec{b}$ and $5 \vec{a}-4 \vec{b}$ are perpendicular. What is the angle between $\vec{a}$ and $\vec{b}$ ?

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

Solution

According to question, $|\vec{a}|=|\vec{c}|=1$ and $(\vec{a}+2 \vec{c}) \cdot(5 \vec{a}-4 \vec{c})=0$ $\Rightarrow 5(\vec{a})^2-4 \vec{a} \cdot \vec{c}+10 \vec{c} \cdot \vec{a}-8|\vec{c}|^2=0$ $\Rightarrow 5+6 \vec{a} \cdot \vec{c}-8=0 \quad \Rightarrow \vec{a} \cdot \vec{c}=\frac{1}{2}$ Now, $\cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}=\frac{1}{2}$ $\Rightarrow \cos \theta=\cos \frac{\pi}{3} \Rightarrow \theta=\frac{\pi}{3}$

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

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