5. Cross Product

b. Cross Product and Triple Product

4. Right Hand Rule

The direction of the cross product, \(\vec u\times\vec v\), is given by the right hand rule. So we first need the definition of the right hand rule.

There are several equivalent descriptions of the right hand rule all involving your right hand.

Extend your pointer finger out in front of you. Extend your middle finger out in front of your palm (not necessarily perpendicular). And extend your thumb perpendicular to the other two. If \(\vec u\) points along your pointer finger and \(\vec v\) points along your middle finger, then by the right hand rule, \(\vec u\times\vec v\) points along your thumb.

Place the fingers of your right hand along \(\hat{u}\) with the palm facing \(\hat{v}\) and your thumb extended perpendicular to the fingers. Rotate your fingers from \(\hat{u}\) to \(\hat{v}\). Then by the right hand rule your thumb points along \(\vec u\times\vec v\).

Hands illustrated and programmed by
Carl Van Huyck
MS in Visualization, TAMU

In the exercise below, click the refresh button to get a new problem.

  Use the hand below to find the direction of the cross product of \(\vec u = \langle\) \(,\) \(,\) \(\rangle\)   and   \(\vec v = \langle\) \(,\) \(,\) \(\rangle\)
When you get the direction correct, the cross product will appear here:

\(\vec u\times\vec v = \langle\) \(,\) \(,\) \(\rangle\)
Check the answer by computing the cross product as a determinant.

To find the direction of \(\vec u\times\vec v\):

  • Orient the pointer finger along \(\vec u\) using the \(\theta\) and \(\phi\) sliders, which give the spherical coordinates of the pointer finger.
  • Then orient the middle finger so \(\vec v\) is in the plane of the pointer and middle fingers, using the \(\alpha\) slider, which gives the angle around the pointer finger.
  • Optionally, align the middle finger with \(\vec v\) using the \(\beta\) slider, which gives the angle between the pointer finger and the middle finger.
  • Then the thumb will point along \(\vec u\times\vec v\).
  • Pointer:
    \(\theta=\)

    \(\phi=\)
  • Middle:
    \(\alpha=\)

    \(\beta=\)

When aligning the pointer finger, it may help to first rotate the plot so that \(\vec u\) points straight out of the paper.

Using the right hand rule verify the direction of the cross product \(\hat{\imath}\times\hat{\jmath}=\hat{k}\).

Using your real hand, put the fingers of your right hand along the positive \(x\)-axis, the \(\hat{\imath}\) direction, with the palm facing the positive \(y\)-axis, the \(\hat{\jmath}\) direction. As you rotate your fingers from the \(x\)-axis to the \(y\)-axis, your thumb is pointing straight up along the positive \(z\)-axis, which is the \(\hat{k}\) direction. So \(\hat{\imath}\times\hat{\jmath}\) points in the \(\hat{k}\) direction, which agrees with the algebraic formula.

Using the graphical hand in the exercise above, type \(\vec u=\hat{\imath}=\langle1,0,0\rangle\) and \(\vec v=\hat{\jmath}=\langle0,1,0\rangle\) in the boxes. Then click on the show button. The hand will automatically rotate so the pointer is along \(\vec u\) and the middle finger is along \(\vec v\). Then the thumb will be along \(\hat{\imath}\times\hat{\jmath}=\langle0,0,1\rangle=\hat{k}\).

Additionally, use the hand above to check: \[ \hat{\jmath}\times\hat{\imath}=-\hat{k} \qquad \hat{\jmath}\times\hat{k}=\hat{\imath} \qquad \hat{k}\times\hat{\imath}=\hat{\jmath} \]

Right Hand Rule using Compass Directions

 

If \(\vec u\) points and \(\vec v\) points , which way does \(\vec u\times\vec v\) point?

Practice this with your real right hand as well.

  • Pointer:
    \(\theta=\)

    \(\phi=\)
  • Middle:
    \(\alpha=\)

    \(\beta=\)

North   East    Northeast    Southeast    Up  

South   West   Northwest   Southwest   Down

Point with your index finger on your right hand. With your middle finger pointing perpendicular to the index, rotate your hand so that your palm is facing . Your thumb is pointing .

© MYMathApps

Supported in part by NSF Grant #1123255