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)

Inclined at an obtuse angle

$|\overrightarrow{a}+\overrightarrow{b}|<|\overrightarrow{a}-\overrightarrow{b}|$
$a^2+b^2+2ab\cos \theta <a^2+b^2-2ab\cos \theta$ where $\theta$ is the included angle between the two vectors.
Hence
$4ab\cos \theta <0$
Since $a,b>0$ (magnitude of a vector is always a positive quantity).
Hence
$\cos \theta <0$
Or
$\theta \in (\frac{\pi }{2},\frac{3\pi }{2})$. Hence $\theta$ is an obtuse angle.