Cross Product and Applications: Learn It 5

Applications of the Cross Product

The cross product appears throughout mathematics, physics, and engineering. Beyond the geometric applications we’ve seen, cross products solve practical problems involving rotation, planes, and forces. Let’s explore these real-world applications.

Testing for Coplanar Vectors

Three vectors are coplanar if they all lie in the same plane. The triple scalar product provides a simple test: if the volume of the parallelepiped formed by three vectors is zero, then the vectors must be coplanar.

coplanar vector test

Three vectors [latex]\mathbf{u}[/latex], [latex]\mathbf{v}[/latex], and [latex]\mathbf{w}[/latex] are coplanar if and only if [latex]\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = 0[/latex].

Use the triple scalar product to show that vectors [latex]{\bf{u}}=\langle2,0,5\rangle[/latex], [latex]{\bf{v}}=\langle2,2,4\rangle[/latex], and [latex]{\bf{w}}=\langle1,-1,3\rangle[/latex] are coplanar—that is, show that these vectors lie in the same plane.

Finding Vectors Orthogonal to Planes

When you need a vector perpendicular to a plane, cross products provide the solution. Any plane containing two non-parallel vectors can be described using their cross product.

Only a single plane can pass through any set of three noncolinear points. Find a vector orthogonal to the plane containing points [latex]P=(9, -3, -2)[/latex], [latex]Q=(1, 3, 0)[/latex], and [latex]R=(-2,5,0)[/latex].

Finding normal vectors to planes is essential in computer graphics (for lighting calculations), engineering (for surface analysis), and physics (for collision detection). The cross product makes this calculation straightforward.

Torque: Force and Rotation

One of the most important applications of cross products is calculating torque—the rotational effect of a force. When you turn a wrench, tighten a screw, or spin a wheel, you’re applying torque.

Torque

 Torque [latex]\tau[/latex] (Greek letter tau) measures the tendency of a force to produce rotation about an axis. For a force [latex]\mathbf{F}[/latex] applied at position [latex]\mathbf{r}[/latex] from the axis of rotation:

[latex]\tau = \mathbf{r} \times \mathbf{F}[/latex]

This figure has a vector r from an “axis of rotation”. At the terminal point of r there is a vector labeled “F”. The angle between r and F is theta.
Figure 5. Torque measures how a force causes an object to rotate.

Understanding Torque:

  • Magnitude: [latex]|\tau| = |\mathbf{r}| \cdot |\mathbf{F}| \sin \theta[/latex] where [latex]\theta[/latex] is the angle between [latex]\mathbf{r}[/latex] and [latex]\mathbf{F}[/latex]
  • Direction: The torque vector points along the axis of rotation (using the right-hand rule)
  • Maximum effect: Torque is greatest when force is applied perpendicular to the radius vector
  • No effect: When force points toward or away from the axis ([latex]\sin 0° = 0[/latex])

Think about using a wrench to tighten a bolt. The torque [latex]\tau[/latex] applied to the bolt depends on how hard we push the wrench (force) and how far up the handle we apply the force (distance). The torque increases with a greater force on the wrench at a greater distance from the bolt. Common units of torque are the newton-meter or foot-pound.

Although torque has the same units as work (N⋅m or ft⋅lb), they’re different concepts. Work involves force along a line of motion, while torque involves rotational effects around an axis.
A bolt is tightened by applying a force of [latex]6[/latex] N to a [latex]0.15[/latex]-m wrench (Figure 6). The angle between the wrench and the force vector is [latex]40^{\small\circ}[/latex]. Find the magnitude of the torque about the center of the bolt. Round the answer to two decimal places.

This figure is the image of an open-end wrench. The length of the wrench is labeled “0.15 m.” The angle the wrench makes with a vertical vector is 40 degrees. The vector is labeled with “6 N.”
Figure 6. Torque describes the twisting action of the wrench.