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Next: The box product. Up: Analytic geometry in the Previous: Dot product.

Cross product.

Definition 1.17   Let $\overrightarrow{u} $ and $\overrightarrow{v} $ be two vectors in the space.
1.
If at least one of them is equal to $\overrightarrow{0} $ or if they are linearly dependent (parallel), then we define: $\overrightarrow{u}\times \overrightarrow{v} = \overrightarrow{0} $.
2.
Suppose that none of the given vectors is equal to $\overrightarrow{0} $ and that these vectors are linearly independent. Denote by $\theta$ the (oriented) angle from $\overrightarrow{u} $ to $\overrightarrow{v} $. The two given vectors determine a plane $\mathcal{P}$; denote by $\overrightarrow{n} $ a vector, orthogonal to the plane $\mathcal{P}$ and verifying the right-hand rule, i.e. if the right-hand thumb points in the direction of $\overrightarrow{n} $, then the fingers curl through the angle from $\overrightarrow{u} $ to $\overrightarrow{v} $. The cross product $\overrightarrow{u}\times \overrightarrow{v} $ is the vector $\overrightarrow{w} $defined by the two following properties:
  • $\overrightarrow{w} $ is parallel to $\overrightarrow{n} $ and points towards the same direction;
  • $\vert \overrightarrow{w} \vert = \vert \overrightarrow{u} \vert \cdot \vert \overrightarrow{v} \vert \cdot \sin \theta$.

Definition 1.18   A triple of vectors which satisfies the right-hand rule is called direct.


  
Figure 6: The cross product of two vectors.
\begin{figure}
\mbox{\epsfig{file=CrossProduct.eps,height=4cm} }
\end{figure}

\fbox{
$\overrightarrow{u} \times \overrightarrow{v} =
\left(\vert \overrightar...
... \vert \overrightarrow{v} \vert \cdot \sin \theta \right)
\overrightarrow{n}$ }

Proposition 1.19       

1.
$\forall \overrightarrow{u} $, $\overrightarrow{u}\times \overrightarrow{u} = \overrightarrow{0} $.
2.
$\forall \overrightarrow{u} ,\overrightarrow{v} $, $\overrightarrow{u}\times \overrightarrow{v} = - \overrightarrow{v}\times \overrightarrow{u} .$
3.
$\forall \overrightarrow{u} ,\overrightarrow{v} ,\overrightarrow{w} $, $\overrightarrow{u}\times (\overrightarrow{v} + \overrightarrow{w} ) =
\overrightarrow{u}\times \overrightarrow{v} + \overrightarrow{u}\times \overrightarrow{w} $.
4.
$\forall \overrightarrow{u} ,\overrightarrow{v} ,\overrightarrow{w} $, $ ( \overrightarrow{u} + \overrightarrow{v} ) \times \overrightarrow{w} ) =
\ove...
...arrow{u}\cdot \overrightarrow{w} + \overrightarrow{v}\times \overrightarrow{w} $.
5.
$\forall \overrightarrow{u} ,\overrightarrow{v} $, $\forall \lambda \in \mathbb{R} $, $\overrightarrow{u}\times \lambda \overrightarrow{v} = \lambda \overrightarrow{u...
...s \overrightarrow{v} = \lambda (\overrightarrow{u}\times \overrightarrow{v} ) .$


  
Figure 7: The cross product of two vectors: anti-commutativity.
\begin{figure}
\mbox{\epsfig{file=CrossProduct2.eps,height=4cm} }
\end{figure}

Remark 1.20       

$\overrightarrow{i}\times \overrightarrow{j} = \overrightarrow{k} $ $\overrightarrow{j}\times \overrightarrow{i} = - \overrightarrow{k} $
$\overrightarrow{j}\times \overrightarrow{k} = \overrightarrow{i} $ $\overrightarrow{k}\times \overrightarrow{j} = - \overrightarrow{i} $
$\overrightarrow{k}\times \overrightarrow{i} = \overrightarrow{j} $ $\overrightarrow{i}\times \overrightarrow{k} = - \overrightarrow{j} $

Choose now a direct (v.s. Def.  1.18) orthonormal basis $\overrightarrow{i} ,\overrightarrow{j} ,\overrightarrow{k} $ for the 3-dimensional space. Denote:
\begin{align*}\overrightarrow{u} &= u_1 \overrightarrow{i} + u_2 \overrightarrow...
...\overrightarrow{i} + v_2 \overrightarrow{j} + v_3 \overrightarrow{k}\end{align*}
Then:

Proposition 1.21 (Determinant Formula)       


\begin{displaymath}\overrightarrow{u}\times \overrightarrow{v} =
\begin{vmatrix}...
...arrow{k}\\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3
\end{vmatrix}\end{displaymath}

Example 1.22        Take $\overrightarrow{u} =2\overrightarrow{i} + 3 \overrightarrow{j} - 2 \overrightarrow{k} $ and $\overrightarrow{v} =\overrightarrow{i} + 2 \overrightarrow{j} + \overrightarrow{k} $. Then:
\begin{align*}\overrightarrow{u}\times \overrightarrow{v} &=
\begin{vmatrix}\ove...
...= 7 \overrightarrow{i} -4 \overrightarrow{j} + \overrightarrow{k} .
\end{align*}

Example 1.23   Find a unit normal vector for the plane $\mathcal{P}$ through the points A(1,2,-1), B(2,0,1) and C(-1,2,4).

The vector $\overrightarrow{AB}\times \overrightarrow{AC} $ is noemal to $\mathcal{P}$; we have:

\begin{displaymath}\overrightarrow{AB}\times \overrightarrow{AC} =\begin{vmatrix...
...verrightarrow{i} -9 \overrightarrow{j} -4 \overrightarrow{k} .
\end{displaymath}

The norm of this vector is equal to:

\begin{displaymath}\left\vert \overrightarrow{AB}\times \overrightarrow{AC}\right\vert
= \sqrt{ (-10)^2 + (-9)^2 + (-4)^2 } = 197.
\end{displaymath}

The plane $\mathcal{P}$ has two unit normal vectors, namely:

\begin{displaymath}\overrightarrow{n_1} = \frac {1}{\sqrt{197}} (10,9,4) \; ; \; \overrightarrow{n_2} = -\frac {1}{\sqrt{197}} (10,9,4).
\end{displaymath}


next up previous contents
Next: The box product. Up: Analytic geometry in the Previous: Dot product.
Noah Dana-Picard
2001-05-30