Scalar multiples, unit vectors, i j k, the vector AB and direction cosines
Answers are for parents and teachers.
1
For A = (2, −3, 5) and B = (−1, 4, 0) find 3A + 2B and A − 3B.
3A + 2B = (6, −9, 15) + (−2, 8, 0) = (4, −1, 15). A − 3B = (2 + 3, −3 − 12, 5 − 0) = (5, −15, 5).
2
Find x, y, z if (2x, y − 1, z²) = (x + 3, 5, 4x).
2x = x + 3 gives x = 3; y − 1 = 5 gives y = 6; z² = 12 gives z = ±2√3.
3
For P(1, −1, 3) and Q(3, 3, −1) find PQ, its norm and the unit vector in the direction of PQ.
PQ = (2, 4, −4), norm √(4 + 16 + 16) = 6, unit vector (1/3, 2/3, −2/3).
4
Find the direction angles of (2, −2, 1).
The norm is 3, so the cosines are 2/3, −2/3, 1/3. The angles are 48.2°, 131.8° and 70.5°.
5
A cable pulls with 63 N in the direction of (4, 4, 7). Find the components of the force and its angle with the z-axis.
The norm of (4, 4, 7) is 9, so F = 63 × (4, 4, 7)/9 = (28, 28, 49) N. The angle with the z-axis is cos⁻¹(7/9) ≈ 38.9°.