Area of a Triangle in R3 in Terms of Vectors

Area of a Triangle in R3 in Terms of Vectors

This post presents a complete derivation of the area of triangle ABC in R3 using vectors alone. The result follows directly from the definition of the cross product and basic algebraic properties. No geometric shortcuts or previously established vector identities are assumed. Every step is obtained from first principles.

Area of a Triangle in R3 in Terms of Vectors

1. Vector Representation of the Triangle

Let A, B and C be points in R3 with position vectors A, B and C. Two sides of triangle ABC are represented by: 

B - A
C - A

If theta is the angle between these vectors, the classical formula for the area of the triangle is: 

Area = (1/2)*|B - A|*|C - A|*sin(theta)

2. Magnitude of the Cross Product

For any vectors U and V in R3, the magnitude of the cross product is: 

|U x V| = |U|*|V|*sin(theta)

Applying this to U = B - A and V = C - A gives: 

| (B - A) x (C - A) | = |B - A|*|C - A|*sin(theta)

Substituting into the area formula yields: 

Area(ABC) = (1/2)*| (B - A) x (C - A) |

This expresses the area entirely through the magnitude of a single cross product. The next stage is to expand this cross product algebraically.

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

Using the distributive property of the cross product: 

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

Since A x A = (0, 0, 0), this simplifies to: 

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

4. Anti-Commutativity

The cross product satisfies the following identities: 

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

Applying these gives: 

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

This expresses the cross product of the side vectors entirely in terms of A, B and C themselves.

5. Final Area Formula

Substituting this expanded form into the earlier area expression produces the final result: 

Area(ABC) = (1/2)*|A x B + B x C + C x A|

This formula expresses the area of triangle ABC using only the position vectors of its vertices. The derivation relies solely on the definition of the cross product, its algebraic properties, and standard magnitude relationships. The result holds for any triangle embedded in R3.

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