Products between vectors

Take two vectors and multiply them. There is no single answer, because the word multiply is underdetermined. Three different questions are hiding inside it. How much do the vectors overlap? What direction is perpendicular to both? How much space do three of them enclose? Each question has its own product, and each product is exactly one of those answers.

Start with the figure. It puts the three products in one place, one step each. Step through it and watch what each one measures: a projection, a shaded area, a signed volume. The panel prints the live value of whichever product is on screen, so you can see it grow and shrink as you drag.

The dot product keeps only the part of $A$ that points where $B$ points. Drop a perpendicular and read the shadow: that is the projection, and its length times the length of $B$ is $A\cdot B = |A||B|\cos\theta$. Two facts fall straight out. At a right angle there is no shadow, so the product is zero however long the vectors are. And when $B = A$ the product becomes $|A|^2$, the length, which every frame agrees on. That invariance is why the answer is a single number.

The cross product answers a different question: which direction is perpendicular to both? Two independent vectors span a plane, and exactly one direction is normal to it. The product is that direction, with the area of the parallelogram as its length: $|A\times B| = |A||B|\sin\theta$. It is an area with an arrow on it. Swap the two vectors and the parallelogram is unchanged, but the arrow flips: $A\times B = -B\times A$. And $A\times A = 0$, because a vector encloses no area with itself.

Now take three vectors. $B\times C$ is perpendicular to the base, and dotting it with $A$ keeps only the height of $A$ above that base. So $A\cdot(B\times C)$ is base area times height: the volume of the box, with a sign for orientation. Here is the payoff. Drag $C$ into the plane of $A$ and $B$ and the height vanishes, so the volume is zero. A vanishing mixed product is exactly the test for coplanarity. The determinant has been measuring volume all along.

A short decision rule for any product you meet. Need how much of one vector lies along another? Take the dot product and get a scalar. Need the direction perpendicular to two? Take the cross product and get a vector. Need the volume three vectors span? Take the mixed product and get a scalar. Units confirm the choice: work must come out as energy, so only a dot product can produce it; angular momentum must come out a vector, so only a cross product can.

So the three products are not three conventions. They are three measurements: overlap, area, volume. Everything so far has operated on numbers attached to points. The next question asks how such quantities change from point to point, which is where the study of fields begins.