Question

Find the angles between the following pairs of lines:
(i) \(\overrightarrow r= 2\hat i-5\hat j+\hat k+ \lambda (3\hat i-2\hat j+6\hat k)\)and

\(\overrightarrow r=7\hat i-6\hat k+\mu(\hat i+2\hat j+2\hat k)\)

(ii) \(\overrightarrow r=3\hat i+\hat j-2\hat k+\lambda(\hat i-\hat j-2\hat k)\) and

\(\overrightarrow r = 2\hat i-\hat j-56\hat k+\mu(3\hat i-5\hat j-4\hat k)\)

Answer 1

(i) Let Q be the angle between the given lines.

The angle between the given pairs of lines is given by cosQ=\(\begin{vmatrix}\frac{\overrightarrow b. \overrightarrow b}{\| \overrightarrow b \mid \mid \overrightarrow b. \|} \end{vmatrix}\)

The given lines are parallel to the vectors, \(\overrightarrow b_1=3\hat i+2\hat j+6\hat k\, and \overrightarrow b_2=\hat i+2\hat j+2\hat k\), respectively,

∴|\(\overrightarrow b_1\)|=\(\sqrt {3^2+2^2+6^2}\) =7

|\(\overrightarrow b_2\)|=\(\sqrt{(1)^2+(2)^2+(2)^2}=3\)

\(\overrightarrow b_1.\overrightarrow b_2\)

=\((3\hat i+2\hat j+6\hat k\,)(\hat i+2\hat j+2\hat k)\)

=31+22+62
=3+4+12
=19
\(\Rightarrow\) cos Q=\(\frac{19}{73}\)
\(\Rightarrow \) Q= \(\cos^{-1}\)\(\bigg(\frac{19}{21}\bigg)\)

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(ii) The given lines are parallel to the vectors \(\overrightarrow b_1=\hat i-\hat j-2\hat k \) and \(\overrightarrow b_2=3\hat i-5\hat j-4\hat k \), respectively,

∴\(\mid \overrightarrow b_1\mid= \sqrt {(1)^2+(-1)^2+(-2)^2}=\sqrt 6\)

\(\mid\overrightarrow b_1 \mid= \sqrt {(3)^2+(-5)^2+(-4)^2}=\sqrt {50}=5\sqrt 2\)

\(\overrightarrow b_1.\overrightarrow b_2\)

=\((\hat i-\hat j-2\hat k )\).\((3\hat i-5\hat j-4\hat k )\)

=1.3-1(-5)-2(-4)

=3+5+8

=16
cos Q=\(\begin{vmatrix}\frac{\overrightarrow b. \overrightarrow b}{\| \overrightarrow b \mid \mid \overrightarrow b. \|} \end{vmatrix}\)

cos Q=\(\frac{16}{\sqrt16.5\sqrt2}\)

cos Q=\(\frac{16}{\sqrt 2.\sqrt3.5\sqrt2}\)

cos Q=\(\frac{16}{10\sqrt3}\)

cos Q=\(\frac{8}{5\sqrt3}\)

Q=cos^-1\(\frac{8}{5\sqrt3}\)