Question

The angle between the straight lines
$\frac{x+1}{2}=\frac{y-2}{5}=\frac{z+3}{4}\mathrm{and}\frac{x-1}{1}=\frac{y+2}{2}=\frac{z-3}{-3}$ is
(a) 45°
(b) 30°
(c) 60°
(d) 90°

Answer 1

(d) 90°

We have

$$\begin{gathered} \frac{x+1}{2}=\frac{y-2}{5}=\frac{z+3}{4} \\ \frac{x-1}{1}=\frac{y+2}{2}=\frac{z-3}{-3} \end{gathered}$$

The direction ratios of the given lines are proportional to 2, 5, 4 and 1, 2, $-$3.

The given lines are parallel to the vectors $\overrightarrow{b_1}=2\hat{i}+5\hat{j}+4\hat{k}\mathrm{and}\overrightarrow{b_2}=\hat{i}+2\hat{j}-3\hat{k}$.

Let $\theta$ be the angle between the given lines.

Now,
$$\begin{gathered} \cos \theta =\frac{\overrightarrow{b_1}.\overrightarrow{b_2}}{\left|\overrightarrow{b_1}\right|\left|\overrightarrow{b_2}\right|} \\ =\frac{\left(2\hat{i}+5\hat{j}+4\hat{k}\right).\left(\hat{i}+2\hat{j}-3\hat{k}\right)}{\sqrt{2^2+5^2+4^2}\sqrt{1^2+2^2+{\left(-3\right)}^2}} \\ =\frac{2+10-12}{\sqrt{45}\sqrt{14}} \\ =0 \\ \Rightarrow \theta =90° \end{gathered}$$