Question

Angle between the vectors $\overline{a}=-\overline{i}-2\overline{j}+\overline{k}$ and $\overline{b}=x\overline{i}+\overline{j}+(x+1)\overline{k}$.

• A. Obtuse angle
• B. Acute angle
• C. Right angle
• D. Depends on x

Answer 1

The correct option is A Obtuse angle

Let $\theta$ be the angle between given vectors

then,

$\cos \theta =\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{|\overrightarrow{a}{|}^2|\overrightarrow{b}{|}^2}$

$\cos \theta =\frac{-x-2+x+1}{\sqrt{1+4+1}\sqrt{x^2+1+(x+1{)}^2}}$

$\cos \theta =\frac{-1}{\sqrt{6}\sqrt{2x^2+2x+1}}$

$\cos \theta =\frac{-1}{\sqrt{6}\sqrt{x^2+x+1}\sqrt{2}}<0$

$\theta$ is an obtuse angle.