The Cross Product

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(See cross product questions below.)

Cross Products

Based mostly on “The Cross Product and Its Properties” in section 1.4 of the textbook.

Questions?

Where does the formula for magnitude of a cross product come from?

This is probably one of those situations where someone realized the formula and worked out a proof, and then the proof that actually appears in textbook was written after the fact to show the reasons why the relationship holds, but not the thinking through which it was originally discovered.

See the textbook for the line-by-line proof to understand the following comments; we talked through some of its highlights. In particular…

The proof starts by looking at what the square of the magnitude of a cross product would be (with no particular motivation for why one would start with the square, that’s one of the ways in which this proof shows the logic nicely but not the original thinking).

To find the square of the magnitude of the cross product, the proof applies the formula for the square of magnitude to the terms of a cross product.

Then the proof expands the squared differences (e.g., (u₂v₃ - v₂u₃)²) in the resulting formula.

The proof regroups terms resulting from that expansion, but also adds and subsequently subtracts some other terms that will be useful later. This is another place where the proof sets itself up for nice simplifications in future steps, but doesn’t necessarily reflect how someone would come up with this step in the first place.

The regrouped terms are then recognized as the magnitude of u squared, the magnitude of v squared, and the squared dot product of u and v.

Replacing the dot product with the product of the magnitudes times the cosine of the angle between the vectors, factoring out the product of magnitudes, and using the identity 1 - cos²Θ = sin²Θ sets up the final result.

So in the example of calculating the magnitude of a cross product, how did the authors know that the angle between the vectors was π/2?

Because the vectors are conveniently aligned with the y and z axes, which are 90 degrees or π/2 radians apart (vectors won’t usually be so convenient, of course)

Key Ideas

The formula for the cross product. (You can remember it either by remembering the basic pattern, or by using the determinant form in the book.)

The formula for the magnitude of the cross product, |u×v| = |u| |v| sinΘ.

The cross product is a vector, so its rules and algebra are similar to those of vectors.

The cross product of two vectors is orthogonal to both of those vectors.

Examples

Warm-Up

Find the cross product ⟨1, 2, 0⟩ × ⟨-3, 0, 1⟩. Just plug these vectors into your favorite cross product formula:

Normals

In 3-dimensional computer graphics, “normal vectors,” i.e., vectors perpendicular to a surface, are central to calculations of how light reflects from that surface. In mirror-like (including partially mirror-like) reflection, light reflects off a surface at the same angle to the normal that it arrived at; in “diffuse” reflection (the sort of “reflection” responsible for the color and brightness of an object without it reflecting other objects), surfaces become brighter as light reaches them more nearly parallel to the normal.

Suppose that at some point on our computer animated snake, the following vectors are tangent to the surface of the snake:

• ⟨3, -1, -1⟩
• ⟨2, 1, 2⟩

Find a vector normal to the surface at this point.

Since the normal should be perpendicular to both tangents, use their cross product as the normal:

Notice that you get different vectors (negatives of each other) according to the order of the tangents in the cross product. One of these normals will point into the snake, the other will point out.

Wrenches

Why, as a general rule, do people not pull on wrenches at any angle besides perpendicular to the handle? And why do they grab the handle as far from the bolt as possible?

Because torque, the strength of a twist, is also a cross product, namely a radial vector from the center of rotation to the point where a force is applied crossed with the force vector. From the formula for magnitude of a cross product, the torque will be maximized by making the radial vector as long as possible, and its angle to the force 90 degrees (so that sinΘ is 1, the maximum value sine can have):

Next

Put together a lot of what we’ve done with vectors to find equations for lines in 3 dimensions.

Please read “Equations for a Line in Space” and “Distance between a Point and a Line” in section 1.5 of the textbook.

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