Question

why we can't calculate work using cross product?

Answer 1

The cross product accumulates interactions between different dimensions. Taking two vectors, we can write every combination of components in a grid:

cross product interaction grid

This completed grid is the outer product, which can be separated into the:

Dot product, the interactions between similar dimensions (x*x y*y, z*z)

Cross product, the interactions between different dimensions (x*y,y*z, z*x, etc.)

The dot product (
→
a
⋅
→
b
) measures similarity because it only accumulates interactions in matching dimensions. It’s a simple calculation with 3 components.

The cross product (written
→
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):



(The “cross product” assumes 3d vectors, but the concept extends to higher dimensions.)



Defining The Cross Product
The dot product represents vector similarity with a single number:

\displaystyle{\text{dot product} = (a_x, a_y, a_z) \cdot (b_x, b_y, b_z) = a_x b_x + a_y b_y + a_z b_z = \|\vec{a}\| \|\vec{b}\| \cos(\theta)}

(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:

\displaystyle{\text{cross product candidate} = \text{amount of difference} = \|\vec{a}\| \|\vec{b}\| \sin(\theta)}

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) 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*y and 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.)

Geometric Interpretation
Two vectors determine a plane, and the cross product points in a direction different from both (source):

cross product vector diagram

Here’s the problem: there’s two perpendicular directions. By convention, we assume a “right-handed system” (source):

cross product right hand rule

If you hold your first two fingers like the diagram shows, your thumb will point in the direction of the cross product. I make sure the orientation is correct by sweeping my first finger from
→
a
to
→
b
. With the direction figured out, the magnitude of the cross product is ‖a‖‖b‖sin(θ), which is proportional to the magnitude of each vector and the “difference percentage” (sine).

The Cross Product For Orthogonal Vectors
To remember the right hand rule, write the xyz order twice: xyzxyz. Next, find the pattern you’re looking for:

xy => z (x cross y is z)
yz => x (y cross z is x; we looped around: y to z to x)
zx => y
Now, xy and yx have opposite signs because they are forward and backward in our xyzxyz setup.

So, without a formula, you should be able to calculate:

\displaystyle{\vec{x} \times \vec{y} = (1, 0, 0) \times (0, 1, 0) = (0, 0, 1) = \vec{z}}

Again, this is because x cross y is positive z in a right-handed coordinate system. I used unit vectors, but we could scale the terms:

\displaystyle{(3, 0, 0) \times (0, 4, 0) = (0, 0, 12)}

Calculating The Cross Product
A single vector can be decomposed into its 3 orthogonal parts: