Question

The angle between the lines
$\overrightarrow{r}=3\overline{i}+2\overline{j}-4\overline{k}+\lambda (\overline{i}+2\overline{j}+2\overline{k})$ and $\overrightarrow{r}=5\overline{i}-2\overline{k}+m(3\overline{i}+2\overline{j}+6\overline{k})$

• A. ${\cos }^{-1}\left(\frac{18}{21}\right)$
• B. ${\cos }^{-1}\left(\frac{19}{21}\right)$
• C. ${\cos }^{-1}\left(\frac{20}{21}\right)$
• D. ${\cos }^{-1}\left(\frac{17}{21}\right)$

Answer 1

The correct option is B ${\cos }^{-1}\left(\frac{19}{21}\right)$

The angle between the given lines is given by the angle between the vectors giving the direction.

$\overrightarrow{a}=\overrightarrow{i}+2\overrightarrow{j}+2\overrightarrow{k}$

$\overrightarrow{b}=3\overrightarrow{i}+2\overrightarrow{j}+6\overrightarrow{k}$

$\cos \theta =\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}$

$\Rightarrow \cos \theta =\frac{(\overrightarrow{i}+2\overrightarrow{j}+2\overrightarrow{k}).(3\overrightarrow{i}+2\overrightarrow{j}+6\overrightarrow{k})}{\sqrt{1+4+4}\times \sqrt{36+4+9}}$

$\Rightarrow \cos \theta =\frac{3+4+12}{21}$

$\Rightarrow \cos \theta =\frac{19}{21}$