Properties of the Cross Product
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 2.4: “The Cross Product” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product |
| Textbook used in class | Stewart, Calculus, Section 12.4: “The Cross Product” (Examples 4, 5) |
Opening Scenario
The scalar triple product combines the dot and cross products to compute the signed volume of a parallelepiped (the 3D analogue of a parallelogram). If three vectors are coplanar, the parallelepiped they span has zero volume -- so the scalar triple product gives a quick coplanarity test.
Quick Reference
Scalar triple product: \[ \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}. \]
Volume of parallelepiped with edges $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$: \[ V = |\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})|. \]
Coplanarity: $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ are coplanar $\iff \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = 0$.
Cyclic property: $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = \mathbf{b}\cdot(\mathbf{c}\times\mathbf{a}) = \mathbf{c}\cdot(\mathbf{a}\times\mathbf{b})$.
Torque: $\boldsymbol{\tau} = \mathbf{r}\times\mathbf{F}$, where $\mathbf{r}$ is the position vector from the pivot to the point of force application.
Key Concepts
1. Volume of a Parallelepiped
Example 1. Find the volume of the parallelepiped with edges $\mathbf{a} = \langle 1, 2, 3\rangle$, $\mathbf{b} = \langle -1, 1, 2\rangle$, $\mathbf{c} = \langle 2, 1, 4\rangle$. (Stewart 12.4, Example 4.)
\[ \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = \begin{vmatrix}1&2&3\\-1&1&2\\2&1&4\end{vmatrix} = 1(4-2) - 2(-4-4) + 3(-1-2) = 2 + 16 - 9 = 9. \]
Volume $= |9| = 9$.
2. Coplanarity Test
Three vectors are coplanar if and only if the parallelepiped they span has zero volume.
Example 2. Are $\mathbf{a} = \langle 1, 4, -7\rangle$, $\mathbf{b} = \langle 2, -1, 4\rangle$, $\mathbf{c} = \langle 0, -9, 18\rangle$ coplanar?
\[ \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = \begin{vmatrix}1&4&-7\\2&-1&4\\0&-9&18\end{vmatrix} = 1(-18+36) - 4(36-0) + (-7)(-18-0) = 18 - 144 + 126 = 0. \]
Boxed answer: Yes, the three vectors are coplanar.
3. Torque
Example 3. A force $\mathbf{F} = \langle 0, 6, -8\rangle$ N acts at a point $P$ with position vector $\mathbf{r} = \langle 3, -1, 5\rangle$ m from a pivot. Find the torque and its magnitude. (Stewart 12.4, Example 5.)
\[ \boldsymbol{\tau} = \mathbf{r}\times\mathbf{F} = \begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\3&-1&5\\0&6&-8\end{vmatrix} = \langle (-1)(-8)-(5)(6), (5)(0)-(3)(-8), (3)(6)-(-1)(0)\rangle = \langle 8-30, 0+24, 18\rangle = \langle -22, 24, 18\rangle. \]
$|\boldsymbol{\tau}| = \sqrt{484 + 576 + 324} = \sqrt{1384} = 2\sqrt{346} \approx 37.2$ N-m.
computing $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})$ as $(\mathbf{a}\cdot\mathbf{b})\times\mathbf{c}$. The scalar triple product groups as $\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})$: first cross product, then dot product. The expression $(\mathbf{a}\cdot\mathbf{b})\times\mathbf{c}$ is the cross product of a scalar with a vector, which is undefined. The parentheses matter.
Common Errors Summary
| Error | Correction |
|---|---|
| Sign error in the $3\times 3$ determinant | Expand carefully: $+a_1 M_{11} - a_2 M_{12} + a_3 M_{13}$ |
| Confusing volume with the signed triple product | Take the absolute value to get the (positive) volume |
Common Misconceptions
the scalar triple product can be computed as $(\mathbf{a} \cdot \mathbf{b}) \times \mathbf{c}$.
This is the concept-image-conflicts-definition error. The scalar triple product is $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$: perform the cross product first, then the dot product. The expression $(\mathbf{a} \cdot \mathbf{b}) \times \mathbf{c}$ requires taking the cross product of a scalar with a vector, which is undefined. In the scalar triple product, the cross product always acts between two vectors, yielding a vector, and the dot product acts last, yielding a scalar.
Leveled Practice
Problem 1. Compute the volume of the parallelepiped with edges $\mathbf{a} = \langle 1, 0, 1\rangle$, $\mathbf{b} = \langle 0, 2, 0\rangle$, $\mathbf{c} = \langle 1, 1, 0\rangle$.
Show answer
$\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = \begin{vmatrix}1&0&1\\0&2&0\\1&1&0\end{vmatrix} = 1(0-0) - 0(0-0) + 1(0-2) = -2$.
Volume $= |-2| = 2$.