Question

Let \( L \) be the line passing through the points \( i\hat{i} - 9\hat{k} \) and \( 7j\hat{k} \), and \( \pi \) be the plane passing through the point \( 6i + j \) and perpendicular to the vector \( i + j + k \). If \( \theta \) is the angle between \( L \) and \( \pi \), then \( \sin \theta = \):

• A. \( \frac{8\sqrt{2}}{15} \)
• B. \( \frac{3\sqrt{3}}{8} \)
• C. \( \frac{7}{13} \)
• D. \( \frac{24}{25} \)

Hint:

To calculate the angle between a line and a plane, use the formula \( \sin \theta = \frac{\left| \mathbf{a} \cdot \mathbf{n} \right|}{\left| \mathbf{a} \right| \left| \mathbf{n} \right|} \), where \( \mathbf{a} \) is the direction vector of the line and \( \mathbf{n} \) is the normal vector of the plane.

Answer 1

The Correct Option is A

We are given that \( L \) is the line passing through the points \( i\hat{i} - 9\hat{k} \) and \( 7j\hat{k} \), and \( \pi \) is the plane passing through the point \( 6i + j \) and perpendicular to the vector \( i + j + k \). The angle between the line and the plane is found using the formula involving the direction ratios of the line and the normal to the plane. We calculate \( \sin \theta \) using the dot product between the direction vector of the line and the normal vector of the plane, leading to the answer \( \frac{8\sqrt{2}}{15} \).