Higher Mathematics · Free original practice · 4 marks
Scalar product and unit vector
Work through one question, use a small hint if needed, then compare your reasoning. Suggested marks are for self-assessment; independent subject review is pending.
For a = (2, −1, 2) and b = (1, 2, 0), show that a and b are perpendicular and find a unit vector in the direction of a.
A small hint
Perpendicular nonzero vectors have scalar product zero.
Worked explanation and suggested marking
a·b = 2 × 1 + (−1) × 2 + 2 × 0 = 0.
Up to 1 mark
The nonzero vectors are therefore perpendicular.
Up to 1 mark
|a| = √(4 + 1 + 4) = 3.
Up to 1 mark
Divide a by 3 to obtain (2/3, −1/3, 2/3).
Up to 1 mark
a·b = 0; unit vector = (2/3, −1/3, 2/3)
What to watch for
- Do not divide by the sum of the components.