Find the angle between the unit vectors b and c if the vectors c-2b and 4b + 5c are perpendicular.

By hypothesis, the vectors (с ̅ – 2 b ̅) and (4 b ̅ + 5 с) are perpendicular, which means that their scalar product is equal to zero:

(c ̅ – 2 b ̅) (4 b ̅ + 5 c ̅) = 5 * c ^ 2 – 6 * (b ̅ c ̅) – 8 * b ^ 2 = 0

The vectors with ̅ and b ̅ are unit vectors, therefore:

c ^ 2 = b ^ 2 = 1; (b ̅ c ̅) = b * c * cosA = cosA;

Here A is the required angle.

Substituting these expressions into the equation for the dot product, we get:

5 – 6 * cosA – 8 = 0

This implies:

cosA = -1/2;

The acute angle A, whose cosine is -1/2, is 120 degrees.