Find the angle between the vectors $2\hat{{i}} - \hat{{j}} + \hat{{k}}$ and $3\hat{{i}} + 4\hat{{j}} - \hat{{k}}$.

$\cos^{-1} \left( \frac{1}{2\sqrt{39}} \right)$

The correct answer is Option (4) → $\cos^{-1} \left( \frac{1}{2\sqrt{39}} \right)$ ##

Let ${a} = 2\hat{{i}} - \hat{{j}} + \hat{{k}}$ and ${b} = 3\hat{{i}} + 4\hat{{j}} - \hat{{k}}$

We know that, angle between two vectors ${a}$ and ${b}$ is given by

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

$[∵(a_1\hat{{i}} + b_1\hat{{j}} + c_1\hat{{k}}) \cdot (a_2\hat{{i}} + b_2\hat{{j}} + c_2\hat{{k}}) = a_1a_2 + b_1b_2 + c_1c_2]$

$= \frac{(2\hat{{i}} - \hat{{j}} + \hat{{k}})(3\hat{{i}} + 4\hat{{j}} - \hat{{k}})}{\sqrt{4 + 1 + 1}\sqrt{9 + 16 + 1}}$

$= \frac{6 - 4 - 1}{\sqrt{6}\sqrt{26}} = \frac{1}{2\sqrt{39}}$

$∴\theta = \cos^{-1}\left( \frac{1}{2\sqrt{39}} \right)$