Question

For non-zero vectors $\overrightarrow{a} $ and $\overrightarrow{b}$ , if $|\overrightarrow{a} +\overrightarrow{b}|<|\overrightarrow{a} -\overrightarrow{b}|,$ then $\overrightarrow{a} $ and $,\overrightarrow{b} $ are

• A. collinear
• B. perpendicular to each other
• C. inclined at an acute angle
• D. inclined at an obtuse angle.

Answer 1

Answer: (D)

Since $\left|\vec{a}+\vec{b}\right| < \left|\vec{a}-\vec{b}\right|$
$\therefore \left|\vec{a}+\vec{b}\right|^{2} < \left|\vec{a}-\vec{b}\right|^{2}$
$\Rightarrow a^{2}+b^{2}+2\,\vec{a} . \vec{b} < a^{2}+b^{2}-2\,\vec{a} . \vec{b}$
$\Rightarrow 4\,\vec{a} .\vec{b} < 0$
$\Rightarrow \vec{a} . \vec{b} < 0$
$\Rightarrow cos\,\theta < 0$
$\Rightarrow $ is obtuse
$(\theta$ is the angle between $\vec{a}, \vec{b})$