SUNY Geneseo Department of Mathematics

The Cross Product

Friday, February 10

Math 223
Spring 2023
Prof. Doug Baldwin

Anything You Want to Talk About?

(No.)

The Cross Product

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

Key Ideas

The formula(s) for computing cross products. See the “Questions” section for discussion of two versions of the formula.

Applications of the cross product include

Given two vectors, finding a third that is orthogonal to both
Finding areas of parallelograms
Via the triple scalar product, finding volumes of parallelepipeds.

Other interesting facts that fit with and support the above include…

As seen in the formula for cross product, the cross product is a vector.
The right hand rule gives you the direction of a cross product without having to calculate it in full: to find the direction of u × v, hold your right hand so the fingers align with u and can curl in the direction of v; your thumb then points in the direction of u × v.
As suggested by the right hand rule, the cross product is not commutative, i.e., u × v ≠ v × u. Instead, cross product is “anti-commutative,” u × v = -(v × u).
The cross product of any vector with itself is always the zero vector, 0.
Cross products lead to some elegant relationships between the standard unit vectors. See the text for examples.
The magnitude of a cross product is the product of the magnitudes of the vectors involved and the sine of the angle between those vectors.

Questions

What is the determinant formula and its connection to cross product?

The determinant formula is basically a way of remembering the formula for a cross product. Both lead to exactly the same computation, but people who are comfortable with determinants don’t have to remember the cross product formula as a separate thing:

Standard and determinant formulas for cross product give same results

The book uses cross products in projections sometimes? How does that work?

The “projection” in these examples is usually as a consequence of the magnitude of a cross product depending in part on the sine of the angle between two vectors. For example, the area of a parallelogram is the length of its base times its height. So if you describe a parallelogram by a vector v along its base and vector u along one of the adjoining sides, the height of the parallelogram turns out to be ||u|| sin a where a is the angle between the vectors — this in turn is a projection of u onto a line perpendicular to v.

Height of parallelogram is length of vector on one side times sine of angle between sides, area is length times height

Try applying this idea to find the area of a parallelogram defined by the vectors ⟨ 1, 0, 2 ⟩ and ⟨ 0, 2, 1 ⟩:

Calculating the area of a parallelogram by taking the magnitude of the cross product of vectors along 2 sides

Notice that you can quickly estimate whether you calculated a cross product right by checking to see if the dot product of your result with each argument is 0.

Problem Set

Problem set 3, on vectors, is available.

Work on it next week and grade it during the week after.

See the handout for the details.

Lines in 3-dimensional space.

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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