Question

The angle between the vectors $\overrightarrow{a}=3\hat{i}-4\hat{j},\overrightarrow{b}=2\hat{i}+3\hat{k}$ is

• A. ${\cos }^{-1}\left(-\frac{1}{3}\right)$
• B. ${\cos }^{-1}\left(\frac{1}{4}\right)$
• C. ${\cos }^{-1}\left(-\frac{1}{2}\right)$
• D. ${\cos }^{-1}\left(-\frac{1}{6}\right)$

Answer 1

The correct option is A

${\cos }^{-1}\left(-\frac{1}{3}\right)$

Explanation for the correct option:

Let $\theta$ be the angle between the two vectors

We know that

$$\begin{gathered} \cos \theta =\frac{\stackrel{\rightharpoonup }{a}.\stackrel{\rightharpoonup }{b}}{\left|\stackrel{\rightharpoonup }{a}\right|\left|\stackrel{\rightharpoonup }{b}\right|} \\ \Rightarrow \cos \theta =\frac{(3\hat{i}-4\hat{j})·(2\hat{i}+3\hat{k})}{\sqrt{9+16}·\sqrt{4+9}} \\ \Rightarrow \cos \theta =\frac{-6}{5\sqrt{13}} \\ \Rightarrow \cos \theta =-0.33 \\ \Rightarrow \theta ={\cos }^{-1}\left(-\frac{1}{3}\right) \end{gathered}$$

Hence, the correct option is (A)