Question

If $\vec{a}$ and $\vec{b}$ are two vectors, such that $\vec{a} \cdot \vec{b}< 0$ and $|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|$, then angle between vectors $\vec{a}$ and $\vec{b}$ is

• A. $\pi$
• B. $\frac{7 \pi}{4}$
• C. $\frac{\pi}{4}$
• D. $\frac{3 \pi}{4}$

Answer 1

Answer: (D)

$|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|$
$|\vec{ a }||\vec{ b }| \cos \pi|=| \vec{ a }|| \vec{ b }|| \sin \theta \mid$
(where $\theta$ is angle between $\vec{a}$ and $\vec{b}$ )
$\Rightarrow |\cos \theta|=|\sin \theta|$
$\Rightarrow \theta=\frac{\pi}{4}$ or $\frac{3 \pi}{4}($ as $0 \leq \theta \leq T \leq \pi)$
But $\vec{a} \cdot \vec{b}<0$
$\therefore \theta=\frac{3 \pi}{4}$