Q.The angle between the line $\vec{r} = 2\hat{i} - 2\hat{j} + 3\hat{k} + \lambda (\hat{i} - \hat{j} + 4\hat{k})$ and the plane $\vec{r} \cdot (\hat{i} + 5\hat{j} + \hat{k}) = 5$ is

\sin^{-1} (\sqrt{3}/\sqrt{19})

\cos^{-1} (\sqrt{3}/\sqrt{19})

\sin^{-1} (3/\sqrt{19})

\cos^{-1} (3/\sqrt{19})

The correct solution involves applying the fundamental concept to derive the final value step by step...

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Q.The angle between the line r⃗=2i^−2j^+3k^+λ(i^−j^+4k^)\vec{r} = 2\hat{i} - 2\hat{j} + 3\hat{k} + \lambda (\hat{i} - \hat{j} + 4\hat{k})r=2i^−2j^​+3k^+λ(i^−j^​+4k^) and the plane r⃗⋅(i^+5j^+k^)=5\vec{r} \cdot (\hat{i} + 5\hat{j} + \hat{k}) = 5r⋅(i^+5j^​+k^)=5 is