The vectors a , b , c satisfy the equation a + b + c = 0 . Show that a $\times$ b = b $\times$ c = c $\times$ a .

taking cross products with a,     M1

a $\times$ (a + b + c) = a $\times$ 0 = 0     A1

using the algebraic properties of vectors and the fact that a $\times$ a = 0 ,     M1

a $\times$ b + a $\times$ c = 0     A1

a $\times$ b = c $\times$ a     AG

taking cross products with b,     M1

b $\times$ (a + b + c) = 0

b $\times$ a + b $\times$ c = 0     A1

a $\times$ b = b $\times$ c     AG

this completes the proof

[6 marks]