Subsection 2.2.1 Definition of the Cross Product
The dot product is an operation that can be performed on any two vectors in \(\RR^n\) for any \(n \geq 1\text{.}\) There are no other conventional products that work in all dimensions. However, there is a special product that works in three dimensions.
Definition 2.2.1.
Let \(u = \begin{pmatrix} u_1 \\ u_2 \\ u_3
\end{pmatrix}\) and \(v = \begin{pmatrix} v_1 \\ v_2 \\
v_3 \end{pmatrix}\) be two vectors in \(\RR^3\text{.}\) The cross product of \(u\) and \(v\) is written \(u \times v\) and defined by the following formula.
\begin{equation*}
u \times v = \begin{pmatrix} u_2v_3 - u_3v_2 \\ u_3v_1 -
u_1v_3 \\ u_1v_2 - u_2v_1 \end{pmatrix}
\end{equation*}
The cross product differs from the dot product in several important ways. First, it produces a new vector in \(\RR^3\text{,}\) not a scalar. For this reason, when working in \(\RR^3\text{,}\) the dot product is often referred to as the scalar product and the cross product as the vector product. Second, the dot product measures, in some sense, the similarity of two vectors. The cross product measures, in some sense, the difference between two vectors. The cross product has a greater magnitude if the vectors are closer to being perpendicular. If \(\theta\) is the angle between \(u\) and \(v\text{,}\) the dot product was expressed in terms of \(\cos \theta\text{.}\) This measures similarity, since \(\cos 0 =
1\text{.}\) There is a similar identity for the cross product.
\begin{equation*}
|u \times v| = |u||v| \sin \theta
\end{equation*}
This identity tells me that the cross product measures difference in direction, since \(\sin 0 = 0\text{.}\) In particular, this tells me that \(|u \times u| = 0\text{,}\) implying that \(u \times u = 0\) (the zero vector is the only vector which has zero length). This is another new and strange property: in this particular multiplication, everything squares to zero. The cross product is obviously very different from the multiplication of scalars, where \(a^2 = 0\) cannot happen unless \(a=0\text{.}\)
Also consider the relationship between \(u\) and \(u \times v\) as calculated through the dot product.
\begin{align*}
u \cdot (u \times v) \amp =
\begin{pmatrix}
u_1 \\ u_2 \\ u_3
\end{pmatrix} \cdot
\begin{pmatrix}
u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 \\ u_1 v_2 - u_2 v_1
\end{pmatrix}\\
\amp = u_1u_2v_3 - u_1u_3v_2 + u_2u_3v_1 - u_2u_1v_3 +
u_3u_1v_2 - u_3u_2v_1 = 0
\end{align*}
A similar calculation shows that \(v \cdot (u \times v) =
0\text{.}\) Since a dot product of two vectors is zero if and only if the vectors are perpendicular, the vector \(v \times u\) is perpendicular to both \(u\) and \(v\text{.}\) This turns out to be a very useful property of the cross product.
Finally, a calculation from the definition shows that \(u
\times v = -(v \times u)\text{.}\) So far, multiplication of scalars and the dot product of vectors have not depended on order. The cross product is one of many products in mathematics which depends on order. If I change the order of the cross product, I introduce a negative sign.
Definition 2.2.2.
Products that do not depend on the order of the factors, such as multiplication of scalars and the dot product of vectors, are called commutative products. Products where changing the order of the factors introduces a negative sign are called anti-commutative products. The cross product is an anti-commutative product. Other products that have neither of these properties are called non-commutative products.
Subsection 2.2.2 Angular Motion
An important application of the cross product is found in describing rotational motion. Linear mechanics describes the motion of an object through space, but rotational mechanics describes the rotation of an object independent of its movement through space. A force on an object can cause both kinds of movement, obviously. The following table summarizes the parallel questions of linear motion and rotational motion in \(\RR^3\text{.}\)
| Linear Motion |
Rotational Motion |
|
|
| Moving in a straight line |
Continual spinning |
| Direction of motion |
Axis of spin |
| Force |
Torque |
| Momentum |
Angular Momentum |
| Mass (resistance to motion) |
Moment of Inertia (resistance to spin) |
| Velocity |
Frequency (Angular Velocity) |
| Acceleration |
Angular Acceleration |
How should I describe torque? If there is a linear force applied to an object which can only rotate around an axis, and if the linear force is applied at a distance \(r\) from the axis, I can think of the force \(F\) and the distance \(r\) as vectors. The torque is then \(\tau = r \times F\text{.}\) Notice that \(|\tau| = |r||F| \sin \theta\text{,}\) indicating that linear force perpendicular to the radius gives the greatest angular acceleration. That makes sense. If \(F\) and \(r\) share a direction, then I am pushing directly along the axis, and no rotation can occur.
The use of cross products in rotational dynamics is ubiquitous. In fluid dynamics, local rotational motion of the fluid result in turbulence, vortices and similar effects. Tornadoes and hurricanes are particularly extreme examples of vortices. All the descriptions of the force and motion of these vortices involve cross products in the vectors describing the fluid.
