Vector (cross) product sheet3-mark answer frameFind a unit vector perpendicular to each of a⃗ + b⃗ and a⃗ − b⃗, where a⃗ = 2î − ĵ + 3k̂ and b⃗ = î + ĵ − k̂.
How to lay out the answer. Follow the frame, then write it in your own words.
- 1 · Write what is given and what is to be found. Given: a⃗ and b⃗. To find: a unit vector perpendicular to both p⃗ = a⃗ + b⃗ and q⃗ = a⃗ − b⃗.
- 2 · Write the formula. A unit vector perpendicular to p⃗ and q⃗ is ±(p⃗ × q⃗)/|p⃗ × q⃗|.
- 3 · Find p⃗ and q⃗. p⃗ = a⃗ + b⃗ = 3î + 0ĵ + 2k̂; q⃗ = a⃗ − b⃗ = î − 2ĵ + 4k̂.
- 4 · Find the cross product.
- p⃗ × q⃗ = |î ĵ k̂; 3 0 2; 1 −2 4|
- = î(0 + 4) − ĵ(12 − 2) + k̂(−6 − 0) = 4î − 10ĵ − 6k̂
- 5 · Find its magnitude. |p⃗ × q⃗| = √(16 + 100 + 36) = √152 = 2√38
- 6 · Check and box the answer. Unit vector = ±(2î − 5ĵ − 3k̂)/√38. Check: (3)(2) + 0 + (2)(−3) = 0 and (1)(2) + (−2)(−5) + (4)(−3) = 0.
Scoring tip: Write both signs (±). Each of the two opposite unit vectors is perpendicular to both given vectors.
