Lecture 3 (Text Section 12.4)
In this lecture we again deﬁne a new product for vectors.
In this case, the multiplication of two vectors yields a third
vector with some very interesting properties and applications.
The cross product of the vectors u and v, represented
symbolically as u × v, is deﬁned to be a vector of magnitude
|u × v| =
where ▯ is the angle (0 ≤ ▯ ≤ ▯) between the vectors and
where the resulting product vector has a direction given by
the right hand rule.
According to the right hand rule, you point the out-
stretched ﬁngers of the right hand in the direction of the vec-
tor u and curl these ﬁngers in the direction of v (in the same
sense as the angle ▯); the direction in which your thumb is
pointing during this operation is the direction of u × v. Example: According to this rule, if your paper corresponds
to the x;y-plane with the positive y-axis pointing to the top
of the page and the positive x-axis pointing to the right of the
page then, i×j is perpendicular to the page pointing upward
and j × i is perpendicular to the page pointing downward.
The ﬁrst results of the deﬁnition of the cross products are
given in the following theorem:
Let u and v be two vectors in R .
1. The vectors u and v are parallel (▯ = 0 or ▯ = ▯) if and
2. If u and v are two sides of a parallelogram then the area
of the parallelogram is The deﬁnition of the cross product does not provide a con-
venient means of computation for component-wise represen-
tations of vectors. This is addressed by the following Theorem:
Let u = ⟨u ;u ;1 ⟩ 2nd v3= ⟨v ;v ;v ⟩, then1 2 3
▯ i j k ▯
u × v = ▯u u u : ▯
▯ 1 2 3 ▯
▯v 1 v 2 v 3 ▯
The determinant in the theorem can be computed in many
ways; perhaps the simplest technique is to write the three
columns of the matrix and add two additional columns by re-
peating the ﬁrst and second columns. Next proceed from left