The Algebra Behind the Cross Product Magnitude

This expansion shows why the expression

|A|²|B|² − (A · B)²

is equal to the squared magnitude of the cross product:

|A × B|²

Let

A = (a₁, a₂, a₃)

and

B = (b₁, b₂, b₃)

Then:

|A|² = a₁² + a₂² + a₃²

|B|² = b₁² + b₂² + b₃²

Therefore:

|A|²|B|²
= (a₁² + a₂² + a₃²)(b₁² + b₂² + b₃²)

Expanding:

|A|²|B|²
= a₁²b₁² + a₁²b₂² + a₁²b₃²
+ a₂²b₁² + a₂²b₂² + a₂²b₃²
+ a₃²b₁² + a₃²b₂² + a₃²b₃²

Now expand the dot product.

A · B = a₁b₁ + a₂b₂ + a₃b₃

So:

(A · B)²
= (a₁b₁ + a₂b₂ + a₃b₃)²

Expanding:

(A · B)²
= a₁²b₁² + a₂²b₂² + a₃²b₃²
+ 2a₁b₁a₂b₂ + 2a₁b₁a₃b₃ + 2a₂b₂a₃b₃

Now subtract:

|A|²|B|² − (A · B)²

The matching diagonal terms cancel:

a₁²b₁²,   a₂²b₂²,   a₃²b₃²

This leaves:

|A|²|B|² − (A · B)²
= a₁²b₂² + a₁²b₃²
+ a₂²b₁² + a₂²b₃²
+ a₃²b₁² + a₃²b₂²
− 2(a₁b₁a₂b₂ + a₁b₁a₃b₃ + a₂b₂a₃b₃)

Comparing with the Cross Product

The cross product is:

A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)

So its squared magnitude is:

|A × B|²
= (a₂b₃ − a₃b₂)²
+ (a₃b₁ − a₁b₃)²
+ (a₁b₂ − a₂b₁)²

Now expand each square.

(a₂b₃ − a₃b₂)²
= a₂²b₃² − 2a₂b₃a₃b₂ + a₃²b₂²

(a₃b₁ − a₁b₃)²
= a₃²b₁² − 2a₃b₁a₁b₃ + a₁²b₃²

(a₁b₂ − a₂b₁)²
= a₁²b₂² − 2a₁b₂a₂b₁ + a₂²b₁²

Adding the three expansions gives:

|A × B|²
= a₁²b₂² + a₁²b₃²
+ a₂²b₁² + a₂²b₃²
+ a₃²b₁² + a₃²b₂²
− 2(a₁b₁a₂b₂ + a₁b₁a₃b₃ + a₂b₂a₃b₃)

This is exactly the same expression as:

|A|²|B|² − (A · B)²

Therefore:

|A × B|² = |A|²|B|² − (A · B)²

Taking the square root gives:

|A × B| = √(|A|²|B|² − (A · B)²)

Main Point

The expression

|A|²|B|² − (A · B)²

is the squared magnitude of the cross product:

|A × B|²

So:

|A × B| = √(|A|²|B|² − (A · B)²)

This is the algebraic bridge between the dot product, the cross product, and the area of a parallelogram.