Find the angle between the unit vectors b and c if the vectors c-2b and 4b + 5c are perpendicular.
June 26, 2021 | education
| 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.

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