Question

Find the angle between the following pairs of lines: 

(i)\(\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}\) and \(\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}\) 

(ii \(\frac{x}{2}=\frac{y}{2}=\frac{z}{1}\) and \(\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}\)

Answer 1

(i) Let \(\overrightarrow b_1\) and \(\overrightarrow b_2\) be the vectors parallel to the pair of lines, \(\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}\) and \(\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}\), respectively.

∴ \(\overrightarrow b_1=2\hat i+5\hat j-3\hat k\) and \(\overrightarrow b_2=-\hat i+8\hat j+4\hat k\)

\(\mid \overrightarrow b_1\mid =\sqrt{(2)^2+(5)^2+(-3)^2}=\sqrt {38}\)

\(\mid \overrightarrow b_2\mid =\sqrt{(-1)^2+(8)^2+(4)^2}=\sqrt {81}=9\)

\(\mid \overrightarrow b_1.\mid \overrightarrow b_2\) = \((2\hat i+5\hat j-3\hat k)\).\((-\hat i+8\hat j+4\hat k)\)

= 2(-1)+5×8+(-3).4
=-2+40-12
=26

The angle, Q, between the given pair of lines is given by the relation,

cos Q=\(\begin{vmatrix}\frac{\overrightarrow b_1.\overrightarrow b_2}{\|\overrightarrow b_1   \| \overrightarrow b_2 \|} \end{vmatrix}\)

\(\Rightarrow\)  cos Q=\(\frac{26}{9\sqrt{38}}\)

\(\Rightarrow\) Q=\(\cos^{-1} \bigg(\frac{26}{9\sqrt{38}}\bigg)\)

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(ii) Let \(\overrightarrow b_1,\overrightarrow b_2\) be the vectors parallel to the given pair of lines, \(\frac{x}{2}=\frac{y}{2}=\frac{z}{1}\) and \(\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}\), respectively.

\(\overrightarrow b_1=2\hat i+2\hat j+\hat k\) and \(\overrightarrow b_2=4\hat i+\hat j+8\hat k\)

∴|\(\overrightarrow b_1\)|=\(\sqrt{(2)^2+(2)^2+(1)^2}=\sqrt9=3\)

|\(\overrightarrow b_2\)|=\(\sqrt{4^2+1^2+8^2}=\sqrt{81}=9\)

\(\overrightarrow b_1.\overrightarrow b_2\)=\((2\hat i+2\hat j+\hat k).\)\((4\hat i+\hat j+8\hat k)\)

=2×4+2×1+1×8
=8+2+8
=18

If Q is the angle between the given pair of lines,

then cos Q=\(\begin{vmatrix}\frac{\overrightarrow b_1.\overrightarrow b_2}{\|\overrightarrow b_1   \| \overrightarrow b_2 \|} \end{vmatrix}\)

\(\Rightarrow\) cos Q=\(\frac{18}{3*9}=\frac{2}{3}\)

\(\Rightarrow\) Q= \(\cos^{-1}\bigg(\frac{2}{3}\bigg)\)