+476 What is a vector cross product, how do I use it, and how is it useful? (I saw it on the calculator)
+102 The cross product accumulates interactions between different dimensions. Taking two vectors, we can write every combination of components in a grid:
The dot product (a⃗ ⋅b⃗ a→⋅b→) measures similarity because it only accumulates interactions in matching dimensions. It’s a simple calculation with 3 components.
The cross product (written a⃗ ×b⃗ a→×b→) has to measure a half-dozen “cross interactions”. The calculation looks complex but the concept is simple: accumulate 6 individual differences for the total.
Instead of thinking “When do I need the cross product?” think “When do I need interactions between different dimensions?”.
Area, for example, is formed by vectors pointing in different directions (the more orthogonal, the better). Indeed, the cross product measures the area spanned by two 3d vectors (source):
Defining the Cross Product
The dot product represents vector similarity with a single number:
(Remember that trig functions are percentages.) Should the cross product (difference between interacting vectors) be a single number too?
Let’s try. Sine is the percentage difference, so we could use:
Unfortunately, we’re missing a lot of detail. x is 100% different from both y and z, but shouldn’t x*y and x*z be different from each other? As Tolstoy wrote, “All happy families are alike; each unhappy family is unhappy in its own way.”
Instead, let’s express these unique differences as a vector:
The size of the cross product is the numeric “amount of difference” (with sin(θ)sin(θ) as the percentage)
The direction of the cross product is based on both inputs: it’s the direction orthogonal to both (i.e., favoring neither)
A vector result represents the x*yand x*z separately, even though y and z are both “100% different” from x.
(Should the dot product be turned into a vector too? Well, we have the inputs and a similarity percentage. There’s no new direction that isn’t available from either input.) 
