ABCDA1B1C1D1 – cubic meter Points F and K are the midpoints of edges C1D1 and BA, respectively.

ABCDA1B1C1D1 – cubic meter Points F and K are the midpoints of edges C1D1 and BA, respectively. Find the cosine of the angle between lines FD and KA1.

Let the side of the cube be d.

Let us carry out a parallel translation of the straight line FD along the edge AD of the cube. We get intersecting lines A1K and AE.

The angle between crossing lines A1K and FD is equal to the angle a.

Consider the face AA1B1B.

Find the length of the straight line AE:

AI = √ (AA1 ^ 2 + AE ^ 2) = √ (d ^ 2 + (d / 2) ^ 2) = (d√5) / 2;

AA1 / AE = sin β;

sin β = d / ((d√5) / 2) = 2 / √5;

A1G = AE / 2 = ((a√5) / 2) / 2 = d√5 / 4 (half the diagonals of the rectangle).

Let’s apply the sine theorem:

sin a / A1I = sin β / A1G;

sin a = (sin β * A1I) / A1G = ((2 / √5) * (d / 2)) / (d√5 / 4) = (d / √5) / ((d√5) / 4 ) = 4/5;

cos a = √ (1 – sin ^ 2 (a)) = √ (1 – (4/5) ^ 2) = √ (9/25) = 3/5 = 0.6.

Answer: 0.6.