Definition 4.1. The Cross Product (First Definition).
\begin{equation}
\mathbf u \times \mathbf v = \left(|\mathbf u||\mathbf v|\sin\theta\right)\mathbf n.\tag{4.1}
\end{equation}
Section 4.1 Definition and Properties of the Cross Product
Given two vectors \(\mathbf u\) and \(\mathbf v\) in \(\mathbb R^3\text{,}\) the cross product \(\mathbf u \times \mathbf v\) is a vector defined as follows.
Note that \(|\mathbf u||\mathbf v||\sin\theta|\) is the length of the cross product \(\mathbf u \times \mathbf v\text{.}\) Since \(\sin(\theta)\) can be negative, we need to include its absolute value.
The vector \(\mathbf n\) is the unit vector that determines the direction of \(\mathbf u \times \mathbf v\) using the right-hand rule. Note that \(\mathbf u \times \mathbf v\) is orthogonal (perpendicular) to both vectors, as shown in Figure 4.2.
From (4.1), it follows that the cross product of two parallel vectors is zero.
Fact 4.5. Parallel Vectors.
\begin{equation}
\mathbf u \parallel \mathbf v \iff \mathbf u \times \mathbf v = \mathbf 0,
\qquad \mathbf u, \mathbf v \neq \mathbf 0.\tag{4.2}
\end{equation}
Theorem 4.6. Properties of the Cross Product.
If \(\mathbf u\text{,}\) \(\mathbf v\text{,}\) and \(\mathbf w\) are any vectors and \(r\text{,}\) \(s\) are scalars, then
-
\(\displaystyle (r\mathbf u) \times (s\mathbf v) = (rs)(\mathbf u \times \mathbf v)\)
-
\(\displaystyle \mathbf u \times (\mathbf v + \mathbf w) = \mathbf u \times \mathbf v + \mathbf u \times \mathbf w\)
-
\(\displaystyle (\mathbf v + \mathbf w) \times \mathbf u = \mathbf v \times \mathbf u + \mathbf w \times \mathbf u\)
-
\(\displaystyle \mathbf v \times \mathbf u = -(\mathbf u \times \mathbf v)\)
-
\(\displaystyle \mathbf 0 \times \mathbf u = \mathbf 0\)
Remark 4.7.
Note that property (4) follows from (4.1) and the fact that \(\sin(-\theta) = -\sin(\theta)\text{.}\)
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