Scalar (dot) product; angle between vectors practice questions.

Determine the value of k for which the vectors u = (2, k, 1) and v = (3, −2, 4) are perpendicular.

• −5
• 2
• 5
• −2

Answer: 5

Perpendicular vectors have zero scalar product: 6 − 2k + 4 = 0, so 10 = 2k and k = 5.

Two vectors satisfy a·b = −6, |a| = 3 and |b| = 4. Determine the angle between a and b.

• 60°
• 120°
• 30°
• 150°

Answer: 120°

cos θ = (a·b)/(|a||b|) = −6/12 = −1/2, so θ = 120°. A negative scalar product always gives an obtuse angle.

The vectors a and b satisfy |a| = 2, |b| = 5, and the angle between them is 60°. Calculate a·b.

• 10
• 5√3
• 0
• 5

Answer: 5

a·b = |a||b|cos θ = 2 × 5 × cos 60° = 10 × 1/2 = 5. Using sin 60° instead of cos 60° gives the trap answer 5√3.

For any two vectors a and b, the expression (a + b)·(a − b) simplifies to which of the following?

• |a|² − |b|²
• |a|² + |b|²
• 2 a·b
• 0

Answer: |a|² − |b|²

Expanding with the distributive property: a·a − a·b + b·a − b·b = |a|² − |b|², since a·b = b·a.

Determine the angle between the vectors p = (1, 2, 2) and q = (2, −2, 1).

• 0°
• 45°
• 90°
• 60°

Answer: 90°

p·q = (1)(2) + (2)(−2) + (2)(1) = 2 − 4 + 2 = 0, so the vectors are perpendicular and the angle is 90°.

Which pair of vectors is perpendicular?

• (1, 2) and (2, 1)
• (2, 3) and (3, −2)
• (4, 1) and (1, 4)
• (3, 3) and (1, 1)

Answer: (2, 3) and (3, −2)

(2, 3)·(3, −2) = 6 − 6 = 0, so these are perpendicular. The other pairs give scalar products 4, 8 and 6.