TS EAMCET · Maths · Three Dimensional Geometry
A plane \(\pi_1\) contains the vectors \(\bar{i}+\bar{j}\) and \(\bar{i}+2 \bar{j}\). Another plane \(\pi_2\) contains the vectors \(2 \bar{i}-\bar{j}\) and \(3 \bar{i}+2 \bar{k} \cdot \bar{a}\) is a vector parallel to the line of intersection of \(\pi_1\) and \(\pi_2\). If the angle \(\theta\) between \(\bar{a}\) and \(\bar{i}-2 \bar{j}+2 \bar{k}\) is acute, then \(\theta=\)
- A \(\frac{\pi}{2}\)
- B \(\frac{\pi}{4}\)
- C \(\operatorname{Cos}^{-1}\left(\frac{4}{3 \sqrt{5}}\right)\)
- D \(\operatorname{Cos}^{-1}\left(\frac{2}{\sqrt{5}}\right)\)
Answer & Solution
Correct Answer
(C) \(\operatorname{Cos}^{-1}\left(\frac{4}{3 \sqrt{5}}\right)\)
Step-by-step Solution
Detailed explanation
\(\bar{n}_1 = (\bar{i}+\bar{j}) \times (\bar{i}+2\bar{j}) = \bar{k}\) \(\bar{n}_2 = (2\bar{i}-\bar{j}) \times (3\bar{i}+2\bar{k}) = -2\bar{i}-4\bar{j}+3\bar{k}\) \(\bar{a} = \bar{n}_1 \times \bar{n}_2 = \bar{k} \times (-2\bar{i}-4\bar{j}+3\bar{k}) = 4\bar{i}-2\bar{j}\)…
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