Multivariate Calculus - Bruce E. Shapiro

Lecture 3
The Cross Product
Definition 3.1 Let v and w be vectors. Then the cross product v×w is a vector
with magnitude
‖v × w‖ = ‖v‖‖w‖ sin θ (3.1)
and whose direction is perpendicular to the plane that contains v and w according
to the right hand rule:
1. Place v, w so that their tails of the vector coincide;
2. curl the fingers of your right hand from through the angle from v to w.
3. Your thumb is pointing in the direction of v × w
The construction of the cross product is illustrated in figure 3.1. Geometrically,
the cross product gives the area of the parallelogram formed by the two vectors, as
illustrated in figure 3.2.
Figure 3.1: Geometry of the cross prouct. v × w is perpendicular to both v and w.
There are a number of different ways to calculate the cross product by components.
For example, suppose that u = (α, β, γ) and v = (a, b, c). Then to calculate
u × v, define the matrix
⎛
0 −γ
⎞
β
U = ⎝ γ 0 −α⎠ (3.2)
−β α 0
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