MHT CET · Maths · Three Dimensional Geometry
The angle between the lines \(\bar{r}=(\hat{\imath}+2 \hat{\jmath}+3 \hat{k})+\lambda(\hat{\imath}+\hat{\jmath}+2 \hat{k})\) and \(\bar{r}=(3 \hat{\imath}+\hat{k})+\lambda^{\prime}(2 \hat{\imath}+\hat{\jmath}-\hat{k}), \lambda, \lambda^{\prime} \in \mathrm{R}\) is
- A \(\cos ^{-1}\left(\frac{1}{6}\right)\)
- B \(\cos ^{-1}\left(\frac{1}{5}\right)\)
- C \(\cos ^{-1}\left(\frac{1}{3}\right)\)
- D \(\cos ^{-1}\left(\frac{2}{3}\right)\)
Answer & Solution
Correct Answer
(A) \(\cos ^{-1}\left(\frac{1}{6}\right)\)
Step-by-step Solution
Detailed explanation
(B)
The direction ratios of the lines are \(1,1,2\) and \(2,1,-1\) and let \(\theta\) be the angle between them
\(\begin{aligned}
\cos \theta &=\left|\frac{(1)(2)+(1)(1)+2(-1)}{\sqrt{1+1+4} \cdot \sqrt{4+1+1}}\right| \\
\cos \theta &=\left|\frac{1}{\sqrt{6} \cdot \sqrt{6}}\right|=\frac{1}{6} \Rightarrow \theta=\cos ^{-1}\left(\frac{1}{6}\right)
\end{aligned}\)
The direction ratios of the lines are \(1,1,2\) and \(2,1,-1\) and let \(\theta\) be the angle between them
\(\begin{aligned}
\cos \theta &=\left|\frac{(1)(2)+(1)(1)+2(-1)}{\sqrt{1+1+4} \cdot \sqrt{4+1+1}}\right| \\
\cos \theta &=\left|\frac{1}{\sqrt{6} \cdot \sqrt{6}}\right|=\frac{1}{6} \Rightarrow \theta=\cos ^{-1}\left(\frac{1}{6}\right)
\end{aligned}\)
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