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Three Dimensional Geometry — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsThree Dimensional Geometry3 Marks
Question
Find the angle between two lines, one of which has direction ratios 2, 2, 1 while the other one is obtained by joining the points (3, 1, 4) and (7, 2, 12).
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Answer

The direction ratios of the line joining the points (3, 1, 4) and (7, 2, 12) are proportional to 4, 1, 8.
Let $\vec{\text{m}}_1$ and $\vec{\text{m}}_2$ be vectors parallel to the lines having direction ratios proportional to 2, 2, 1 and 4, 1, 8.
Now,
$\vec{\text{b}}_1=2\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}$
$\vec{\text{b}}_2=4\hat{\text{i}}+\hat{\text{j}}+8\hat{\text{k}}$
If $\theta$ is the angle between the given lines, then
$\cos\theta=\frac{\vec{\text{m}}_1.\vec{\text{m}}_2}{|\vec{\text{m}}_1||\vec{\text{m}}_2|}$
$=\frac{\big(2\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}\big).\big(4\hat{\text{i}}+\hat{\text{j}}+8\hat{\text{k}}\big)}{\sqrt{2^2+2^2+1^2}\sqrt{4^2+1^2+8^2}}$
$=\frac{8+2+8}{3\times9}$
$=\frac{2}{3}$
$\Rightarrow\theta=\cos^{-1}\big(\frac{2}{3}\big)$

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