Question

(a) Find an equation of a plane ( P ) that passes through the origin, contains the line ( langle 0, t, 0rangle ), and makes an angle of 5.145 with the plane ( z=0 )

(b) Find

Answer 1

To find an equation of a plane that passes through the origin, contains a given line, and makes an angle with another plane, follow these steps: find a vector perpendicular to both the line and the other plane, normalize the vector, and write the equation of the plane using the normalized vector and the origin's coordinates.

Step-by-step explanation:

To find an equation of a plane that passes through the origin, contains the line ⟨0, t, 0⟩, and makes an angle of 5.145 degrees with the plane z=0, we can use the following steps:

1. Find a vector that is perpendicular to both the given line and the z=0 plane.
2. Normalize the vector from step 1 to find the direction of the plane.
3. Write the equation of the plane using the direction vector and the coordinates of the origin.

Step 1: Find a vector perpendicular to both the given line and the z=0 plane. Since the line is parallel to the y-axis, any vector with nonzero x and z components will be perpendicular to the line. Let's use the vector ⟨1, 0, 1⟩.

Step 2: Normalize the vector from step 1 to find the direction of the plane. The normalized vector is ⟨1/sqrt(2), 0, 1/sqrt(2)⟩.

Step 3: Write the equation of the plane using the direction vector and the coordinates of the origin. The equation of the plane is 1/sqrt(2)x + 0y + 1/sqrt(2)z = 0.