The negative of a vector a, written −a, has the same magnitude as a but points in exactly the opposite direction. If a=AB (from point A to point B), then −a=BA.
Worked example: a scalar triple product identity
For any three vectors a,b,c, prove that [a+bb+cc+a]=2[abc], where [xyz]=x⋅(y×z) is the scalar triple product.
Solution: Expand the cross product first:
(b+c)×(c+a)=b×c+b×a+c×a
(the c×c term vanishes). Dotting with a+b and dropping every term where a vector is dotted with a cross product containing itself (always zero):
(a+b)⋅(b×c+b×a+c×a)=a⋅(b×c)+b⋅(c×a)
The scalar triple product is cyclic, so b⋅(c×a)=a⋅(b×c). Both terms are equal, so the sum is 2[abc], as required.
Worked example: unit vectors summing to zero
If a,b,c are unit vectors with a+b+c=0, find a⋅b+b⋅c+c⋅a.
Solution: Square the magnitude of the sum, which is zero:
∣a+b+c∣2=a⋅a+b⋅b+c⋅c+2(a⋅b+b⋅c+c⋅a)=0
Since each is a unit vector, a⋅a=b⋅b=c⋅c=1:
3+2(a⋅b+b⋅c+c⋅a)=0⇒a⋅b+b⋅c+c⋅a=−23
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Once you can add and scale vectors, finding a unit vector in a given direction is the natural next step.