202 4 Geometry and Trigonometry
product of the magnitudes of the vectors and the cosine of the angle between them. A
dot product equal to zero tells us that the vectors are orthogonal. The cross-product of
two vectors is a vector orthogonal to the two vectors and of magnitude equal to the area
of the parallelogram they generate. The orientation of the cross-product is determined
by the right-hand rule: place your hand so that you can bend your palm from the first
vector to the second, and your thumb will point in the direction of the cross-product. A
cross-product equal to zero tells us that the vectors are parallel (although they might point
in opposite directions).
The dot and cross-products are distributive with respect to sum; the dot product is com-
mutative, while the cross-product is not. For the three-dimensional vectors−→u,−→v,−→w,
the number−→u·(−→v×−→w)is the volume taken with sign of the parallelepiped constructed
with the vectors as edges. The sign is positive if the three vectors determine a frame
that is oriented the same way as the orthogonal frame of the three coordinate axes, and
negative otherwise. Equivalently,−→u·(−→v×−→w)is the determinant with the coordinates
of the three vectors as rows.
A useful computational tool is thecab-bacidentity:
−→a ×(−→b ×−→c)=(−→c ·−→a)−→b −(−→b ·−→a)−→c.
The quickest way to prove it is to check it for−→a,−→
b,−→cchosen among the three unit
vectors parallel to the coordinate axes
−→
i,−→
j, and−→
k,and then use the distributivity of
the cross-product with respect to addition. Here is an easy application of this identity.
Example.Prove that for any vectors−→a,
−→
b,−→c,−→
d,(−→a ×−→
b)×(−→c ×−→
d)=(−→a ·(−→
b ×−→
d))−→c −(−→a ·(−→
b ×−→c))−→
d.Solution.We have
(−→a ×−→
b)×(−→c ×−→
d)=(−→
d ·(−→a ×−→
b))−→c −(−→c ·(−→a ×−→
b))−→
d
=(−→a ·(−→
b ×−→
d))−→c −(−→a ·(−→
b ×−→c))−→
d.In the computation we used the equality−→u ·(−→v ×−→w)=−→w·(−→u ×−→v), which is
straightforward if we write these as determinants.
Let us briefly point out a fundamental algebraic property of the cross-product. Denote
by so( 3 )the set of 3×3 matricesAsatisfyingA+At=O 3 endowed with the operation
[A, B]=AB−BA.
Theorem.The map
(a 1 ,a 2 ,a 3 )→⎛
⎝
0 −a 1 −a 2
a 1 0 −a 3
a 2 a 3 0