Given any two non-zero vectors a and b , show that $|$a $\times$ b${|^2}$ = $|$a${|^2}$$|$b${|^2}$ – (a $\cdot$ b)$^2$.

METHOD 1

Use of $|$a $\times$ b$|$ = $|$a$|$$|$b$|$$\sin \theta$     (M1)

$|$a $\times$ b${|^2}$ = $|$a${|^2}$$|$b${|^2}$${\sin ^2}\theta$     (A1)

Note: Only one of the first two marks can be implied.

= $|$a${|^2}$$|$b${|^2}$$(1 - {\cos ^2}\theta )$     A1

= $|$a${|^2}$$|$b${|^2}$ – $|$a${|^2}$$|$b${|^2}$${\cos ^2}\theta$     (A1)

= $|$a${|^2}$$|$b${|^2}$ – $\left( | \right.$a$|$$|$b$|$${\left. {\cos \theta } \right)^2}$     (A1)

Note: Only one of the above two A1 marks can be implied.

= $|$a${|^2}$$|$b${|^2}$ – (a $\cdot$ b)$^2$     A1

Hence LHS = RHS     AG     N0

[6 marks] 

METHOD 2

Use of a $\cdot$ b = $|$a$|$$|$b$|$$\cos \theta$     (M1)

$|$a${|^2}$$|$b${|^2}$ – (a $\cdot$ b)$^2$ = $|$a${|^2}$$|$b${|^2}$ – $\left( | \right.$a$|$$|$b$|$${\left. {\cos \theta } \right)^2}$     (A1)

= $|$a${|^2}$$|$b${|^2}$ –  $|$a${|^2}$$|$b${|^2}$ ${\cos ^2}\theta$     (A1)

Note: Only one of the above two A1 marks can be implied.

= $|$a${|^2}$$|$b${|^2}$$(1 - {\cos ^2}\theta )$     A1

= $|$a${|^2}$$|$b${|^2}$${\sin ^2}\theta$     A1

= $|$a $\times$ b${|^2}$     A1

Hence LHS = RHS     AG     N0 

Notes: Candidates who independently correctly simplify both sides and show that LHS = RHS should be awarded full marks.

If the candidate starts off with expression that they are trying to prove and concludes that ${\sin ^2}\theta  = (1 - {\cos ^2}\theta )$ award M1A1A1A1A0A0.

If the candidate uses two general 3D vectors and explicitly finds the expressions correctly award full marks. Use of 2D vectors gains a maximum of 2 marks.

If two specific vectors are used no marks are gained.

[6 marks]

Those candidates who chose to use the trigonometric version of Pythagoras’ Theorem were generally successful, although a minority were unconvincing in their reasoning. Some candidates adopted a full component approach, but often seemed to lose track of what they were trying to prove. A few candidates used 2-dimensional vectors or specific rather than general vectors.