Find the angle between the pairs of lines: ...

NCERT Class XII Mathematics Chapter - - Solutions

Question : 15 of 78

Marks: +1, -0

Find the angle between the pairs of lines:

\vec{r}=2\hat{i}-5\hat{j}+\hat{k}

+\lambda(3\hat{i}+2\hat{j}+6\hat{k})

and

\vec{r}=7\hat{i}-6\hat{k}

+\mu(\hat{i}+2\hat{j}+2\hat{k})

Solution:

Here,

\vec{b_1}=3\hat{i}+2\hat{j}+6\hat{k}

and

\vec{b_2}=\hat{i}+2\hat{j}+2\hat{k}

Let θ be the angle between the given lines, then

\cos\theta = \left| \frac{ \vec{b_1} \cdot \vec{b_2} }{ |\vec{b_1}| |\vec{b_2}| } \right|

= \frac{ \left| 3\times 1+2\times 2+6\times 2 \right| }{ \sqrt{3^2+2^2+6^2} \sqrt{1^2+2^2+2^2} }

= \frac{19}{21}

\Rightarrow \theta = \cos^{-1}\left( \frac{19}{21} \right)

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