Question

The angle between the planes \( \vec{r} \cdot (12\hat{i} + 4\hat{j} - 3\hat{k}) = 5 \) and \( \vec{r} \cdot (5\hat{i} + 3\hat{j} + 4\hat{k}) = 7 \) is:

• A. \( \cos^{-1}\left( \frac{12}{13} \right) \)
• B. \( \cos^{-1}\left( \frac{6\sqrt{2}}{13} \right) \)
• C. \( \cos^{-1}\left( \frac{3\sqrt{2}}{13} \right) \)
• D. \( \cos^{-1}\left( \frac{6}{13} \right) \)

Hint:

The angle between two planes is based on the angle between their normal vectors. Use the dot product and magnitudes to compute the cosine of the angle.

Answer 1

The Correct Option is B

The angle between two planes is given by the formula: \[ \cos \theta = \frac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1| |\vec{n}_2|}, \] where \( \vec{n}_1 \) and \( \vec{n}_2 \) are the normal vectors to the planes. The normal vector to the first plane is \( \vec{n}_1 = 12\hat{i} + 4\hat{j} - 3\hat{k} \), and the normal vector to the second plane is \( \vec{n}_2 = 5\hat{i} + 3\hat{j} + 4\hat{k} \). Now, calculate the dot product \( \vec{n}_1 \cdot \vec{n}_2 \) and the magnitudes of the normal vectors \( |\vec{n}_1| \) and \( |\vec{n}_2| \). After performing the calculations, we find that the cosine of the angle is: \[ \cos \theta = \frac{6\sqrt{2}}{13}. \] Thus, the correct answer is \( \cos^{-1}\left( \frac{6\sqrt{2}}{13} \right) \).