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Section 1.18 The Cross Product

The cross product is fundamentally a directed area. The magnitude of the cross product is defined to be the area of the parallelogram whose sides are the two vectors in the cross product.

Figure 1.18.1. The cross product as a directed area.

In the figure above, the height of the parallelogram is \(|\ww|\sin\theta\text{,}\) so its area is

\begin{equation} |\vv\times\ww| = |\vv||\ww|\sin\theta\label{crossmagnitude}\tag{1.18.1} \end{equation}

which is therefore the magnitude of the cross product.

An immediate consequence of Equation (1.18.1) is that, if two vectors are parallel, their cross product is zero,

\begin{equation} \vv\parallel\ww \Longleftrightarrow \vv\times\ww=\zero .\tag{1.18.2} \end{equation}

The direction of the cross product is given by the right-hand rule: Point the fingers of your right hand along the first vector (\(\vv\)), and curl your fingers toward the second vector (\(\ww\)). You may have to flip your hand over to make this work. Now stick out your thumb; that is the direction of \(\vv\times\ww\text{.}\) In the example shown above, \(\vv\times\ww\) points out of the page. The right-hand rule implies that

\begin{equation} \ww\times\vv = -\vv\times\ww\tag{1.18.3} \end{equation}

as you should verify for yourself by suitably positioning your hand. Thus, the cross product is not commutative. 1  Another important property of the cross product is that the cross product of a vector with itself is zero,

\begin{equation} \vv\times\vv = \zero\tag{1.18.4} \end{equation}

which follows from any of the preceding three equations.

The cross product also fails to be associative, since for example \(\xhat\times(\xhat\times\yhat)=-\yhat\) but \((\xhat\times\xhat)\times\yhat=\zero\text{.}\)

In terms of the standard orthonormal basis, the geometric formula quickly yields

\begin{align*} \xhat\times\yhat \amp = \zhat ,\\ \yhat\times\zhat \amp = \xhat ,\\ \zhat\times\xhat \amp = \yhat . \end{align*}

This cyclic nature of the cross product can be emphasized by abbreviating this multiplication table as shown in the figure below. 2 

This is really the multiplication table for the unit imaginary quaternions, a number system which generalizes the familiar complex numbers. Quaternions predate vector analysis, which borrowed the \(i\text{,}\) \(j\text{,}\) \(k\) notation for the rectangular basis vectors, which are often written as \(\ii\text{,}\) \(\jj\text{,}\) \(\kk\text{.}\) Here, we have adopted instead the more logical names \(\xhat\text{,}\) \(\yhat\text{,}\) \(\zhat\text{.}\)
Figure 1.18.2. The cross product multiplication table.

Products in the direction of the arrow get a plus sign; products against the arrow get a minus sign.

Using an orthonormal basis such as \(\{\xhat,\yhat,\zhat\}\text{,}\) the geometric formula reduces to the standard component form of the cross product.  3  If \(\vv=v_x\,\xhat+v_y\,\yhat+v_z\,\zhat\) and \(\ww=w_x\,\xhat+w_y\,\yhat+w_z\,\zhat\text{,}\) then

\begin{align} \vv\times\ww \amp = (v_x\,\xhat+v_y\,\yhat+v_z\,\zhat) \times (w_x\,\xhat+w_y\,\yhat+w_z\,\zhat)\notag\\ \amp = (v_y w_z - v_z w_y)\,\xhat + (v_z w_x - v_x w_z) \,\yhat + (v_x w_y - v_y w_x)\,\zhat\label{eq-crossalg}\tag{1.18.5} \end{align}

which is often written as the symbolic determinant

\begin{equation} \vv\times\ww = \left| \matrix{\xhat\amp \yhat\amp \zhat\cr v_x\amp v_y\amp v_z\cr w_x\amp w_y\amp w_z\cr} \right| .\label{eq-crossdet}\tag{1.18.6} \end{equation}
This argument uses the distributive property, which must be proved geometrically if one starts with  (1.18.1) and the right-hand rule. This is straightforward in two dimensions, but somewhat more difficult in three dimensions. As with the dot product, see our online article.

We encourage you to use (1.18.6), rather than simply memorizing (1.18.5). We also encourage you to compute the determinant as described below, rather than using minors; this tends to minimize sign errors. A \(3\times3\) determinant can be computed in the form

Figure 1.18.3. The determinant.

where one multiplies the terms along each diagonal line, subtracting the products obtained along lines going down to the left from those along lines going down to the right. While this method works only for (\(2\times2\) and) \(3\times3\) determinants, it emphasizes the cyclic nature of the cross product.

Another important skill is knowing when not to use a determinant at all. For simple cross products, such as \((\xhat+3\,\yhat)\times\zhat\text{,}\) it is easier to use the multiplication table directly.

It is also worth pointing out that the multiplication table and the determinant method generalize naturally to any (right-handed) orthonormal basis; all that is needed is to replace the rectangular basis \(\{\xhat,\yhat,\zhat\}\) by the one being used (in the right order!). For example, in cylindrical coordinates, not only is

\begin{equation} \shat\times\phat = \zhat\tag{1.18.7} \end{equation}

(and cyclic permutations), but cross products can be computed as

\begin{equation} \vv\times\ww = \left| \matrix{\shat\amp \phat\amp \zhat\cr v_s\amp v_\phi\amp v_z\cr w_s\amp w_\phi\amp w_z\cr} \right|\tag{1.18.8} \end{equation}

where of course \(\vv=v_s\,\shat+v_\phi\,\phat+v_z\,\zhat\) and similarly for \(\ww\text{.}\)

A good problem emphasizing the geometry of the cross product is to find the area of the triangle formed by connecting the tips of the vectors \(\xhat\text{,}\) \(\yhat\text{,}\) \(\zhat\) (whose base is at the origin).