Date | May 2021 | Marks available | 4 | Reference code | 21M.1.AHL.TZ2.5 |
Level | Additional Higher Level | Paper | Paper 1 (without calculator) | Time zone | Time zone 2 |
Command term | Show that | Question number | 5 | Adapted from | N/A |
Question
Given any two non-zero vectors, a and b, show that |a×b|2=|a|2|b|2-(a·b)2.
Markscheme
METHOD 1
use of |a×b|=|a||b|sin θ on the LHS (M1)
|a×b|2=|a|2|b|2 sin2 θ A1
=|a|2|b|2 (1-cos2 θ) M1
=|a|2|b|2 -|a|2|b|2 cos2 θ OR =|a|2|b|2 -(|a||b| cos θ)2 A1
=|a|2|b|2-(a·b)2 AG
METHOD 2
use of a·b=|a||b|cos θ on the RHS (M1)
=|a|2|b|2 -|a|2|b|2 cos2 θ A1
=|a|2|b|2 (1-cos2 θ) M1
=|a|2|b|2 sin2 θ OR =(|a||b| sin θ)2 A1
=|a×b|2 AG
Note: If candidates attempt this question using cartesian vectors, e.g
a=(a1a2a3) and b=(b1b2b3),
award full marks if fully developed solutions are seen.
Otherwise award no marks.
[4 marks]