Vectors in space
Answers are for parents and teachers.
1
For A(2, −3, 1), B(−1, 1, 13) find the coordinates and length of AB.
AB = (−3, 4, 12), |AB| = √(9 + 16 + 144) = 13.
2
Find the cosine of the angle between a(1, 1, 0) and b(0, 1, 1), and the angle.
a · b = 1, |a| = |b| = √2, cos φ = 1/2, so φ = 60°.
3
Find m for which a(m, 2, −3) and b(4, m, 2) are perpendicular.
4m + 2m − 6 = 0 ⇒ m = 1.
4
Are a(3, −6, 9) and b(−1, 2, −3) collinear? If so, find λ in a = λb.
Yes: the ratios 3/(−1) = −6/2 = 9/(−3) = −3, so a = −3b.
5
If |a| = 3, |b| = 4 and the angle between them is 60°, find a · b and |a + b|².
a · b = 12 · 1/2 = 6. |a + b|² = 9 + 2 · 6 + 16 = 37.