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 $\vec{a},\vec{b})$