Question

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

• A. (A) is obtuse angle
• B. (B) is acute angle
• C. (C) is ${90}^{\circ }$
• D. (D) depends on $x$

Answer 1

Answer: (B), (D)

(B) is acute angle
(D) depends on $x$

Angle between $\overrightarrow{a}$ and $\overrightarrow{b}$,
$\cos \theta =\frac{\overrightarrow{a}.\overrightarrow{b}}{ab}=\frac{-x+2+x+1}{\sqrt{1+4+1}\sqrt{x^2+1^2+(x+1{)}^2}}$ $=\frac{3}{\sqrt{6\left[\begin{array}{c}{x}^2+1+(x+1{)}^2\end{array}\right]}}>0$
Hence $\theta$ is acute angle and dependent on $x$.