In the cube ABCDA1B1C1D1 all edges are equal to 3. A point K is marked on its edge BB1 so that KB

In the cube ABCDA1B1C1D1 all edges are equal to 3. A point K is marked on its edge BB1 so that KB = 2. The plane α is drawn through the points K and C1, parallel to the straight line BD1. a) Prove that the plane α passes through the midpoint of the edge A1B1. b) Find the angle of inclination of the plane α to the plane of the face BB1C1C.

Let’s use the plane equation: – ax + by + cz + d = 0.
Let us give the coordinates of points C1 (0; 4; 4), K (4; 4; 3).
Substituting the coordinates of these points into the equation, we get a system of three equations:
4b + 4c + d = 0,
4a + 4b + 3c + d = 0,
4a + 8 / 3b + 4c + d = 0.
Subtract and get the following equivalent system of equations:
– 4a + c = 0,     a = 1 / 4c,
– 4a + 4 / 3b = 0,    b = 3a = 3 / 4c,
4 / 3b – c = 0,        b = 3 / 4c.
We get the equation of the plane: x + 3y + 4z + d = 0.
Let’s find the cosine of the angle between the planes PC1K and BB1C:
cos B = 3 / √26.
Thus, the required PНB = arctan √17 / 3.
Answer: PНB = arctan √17 / 3.