The scalar product, angles between vectors and components
Answers are for parents and teachers.
1
‖A‖ = 6, ‖B‖ = 4 and A · B = −12. Find the angle between A and B, and ‖A + B‖.
cos θ = −12/24 = −1/2, so θ = 120°. ‖A + B‖² = 36 − 24 + 16 = 28, so ‖A + B‖ = 2√7.
2
ABCD is a square of side 5. Find AB · AC, AB · BC and AC · BD.
AB · AC = 5 × 5√2 × cos 45° = 25. AB · BC = 0. The diagonals of a square are perpendicular, so AC · BD = 0.
3
Find the angle between (2, −2, 1) and (−2, 1, 2).
A · B = −4 − 2 + 2 = −4 and both norms are 3, so cos θ = −4/9 and θ ≈ 116.4°.
4
For which k is (5, k, −1) perpendicular to (2, 3, k)?
10 + 3k − k = 0, so k = −5. Check: (5, −5, −1) · (2, 3, −5) = 10 − 15 + 5 = 0.
5
A force F = (9, 3, 3) N acts along the line with direction (1, 2, 2). Find the component of F in that direction and the projection vector.
F · d = 9 + 6 + 6 = 21 and ‖d‖ = 3, so the component is 7 N. The projection vector is 7 × (1, 2, 2)/3 = (7/3, 14/3, 14/3).