Full Coordinate Derivation of (B - A) x (C - A) in R3

Full Coordinate Derivation of (B - A) x (C - A) in R3

This derivation shows every algebraic step involved in expanding the cross product (B - A) x (C - A) using only coordinates. No vector identities are assumed in advance. All identities that appear at the end arise directly from the coordinate formula and elementary algebra. This method provides complete transparency and is the foundation for many geometric and analytic results involving the cross product.

1. Vectors and Cross Product Formula

A = (a1, a2, a3)
B = (b1, b2, b3)
C = (c1, c2, c3)

For vectors U = (u1, u2, u3) and V = (v1, v2, v3), the cross product is defined in coordinates by: 

U x V =
(
 u2*v3 - u3*v2,
 u3*v1 - u1*v3,
 u1*v2 - u2*v1
)

This is the only formula used. Every identity later in the derivation follows from substituting coordinates into this definition.

2. Basic Cross Products Needed for the Expansion

A x A

A x A =
(
 a2*a3 - a3*a2,
 a3*a1 - a1*a3,
 a1*a2 - a2*a1
)
= (0, 0, 0)

A vector crossed with itself is zero because each component is of the form x*y - y*x = 0.

A x C

A x C =
(
 a2*c3 - a3*c2,
 a3*c1 - a1*c3,
 a1*c2 - a2*c1
)

B x A

B x A =
(
 b2*a3 - b3*a2,
 b3*a1 - b1*a3,
 b1*a2 - b2*a1
)

B x C

B x C =
(
 b2*c3 - b3*c2,
 b3*c1 - b1*c3,
 b1*c2 - b2*c1
)

These expressions will appear naturally inside the expansion of (B - A) x (C - A).

3. Sign Reversals

From the coordinate expressions, the following hold: 

A x B = -(B x A)
C x A = -(A x C)
A x A = (0, 0, 0)

These identities will be used at the end to simplify the final expression.

4. Differences B - A and C - A

B - A = (b1 - a1, b2 - a2, b3 - a3)
C - A = (c1 - a1, c2 - a2, c3 - a3)

The goal is to expand (B - A) x (C - A) term by term using the coordinate definition.

5. Expansion of (B - A) x (C - A)

Let U = B - A and V = C - A. Then: 

(B - A) x (C - A) =
(
 U2*V3 - U3*V2,
 U3*V1 - U1*V3,
 U1*V2 - U2*V1
)

First Component

(b2 - a2)*(c3 - a3) - (b3 - a3)*(c2 - a2)
Expand both products: 
b2*c3 - b2*a3 - a2*c3 + a2*a3
- (b3*c2 - b3*a2 - a3*c2 + a3*a2)
Group into known cross product components: 
= (B x C)_1 - (B x A)_1 - (A x C)_1 + (A x A)_1

Second Component

(b3 - a3)*(c1 - a1) - (b1 - a1)*(c3 - a3)
Expand: 
b3*c1 - b3*a1 - a3*c1 + a3*a1
- (b1*c3 - b1*a3 - a1*c3 + a1*a3)
Group: 
= (B x C)_2 - (B x A)_2 - (A x C)_2 + (A x A)_2

Third Component

(b1 - a1)*(c2 - a2) - (b2 - a2)*(c1 - a1)
Expand: 
b1*c2 - b1*a2 - a1*c2 + a1*a2
- (b2*c1 - b2*a1 - a2*c1 + a2*a1)
Group: 
= (B x C)_3 - (B x A)_3 - (A x C)_3 + (A x A)_3

Vector Form

(B - A) x (C - A)
= B x C - B x A - A x C + A x A

6. Substitution of Known Identities

Using:

A x B = -(B x A)
C x A = -(A x C)
A x A = (0, 0, 0)
Substitute into the expression: 
(B - A) x (C - A)
= B x C + A x B + C x A
Reorder: 
(B - A) x (C - A)
= A x B + B x C + C x A

Final Result

(B - A) x (C - A) = A x B + B x C + C x A

This derivation follows directly from the coordinate definition of the cross product. Every term in the final expression is accounted for by explicit expansion. No geometric arguments or prior vector identities were used. This establishes the identity purely from algebraic first principles.

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