Vector Cross Product Distributes over Addition/Proof 2
Theorem
The vector cross product is distributive over addition.
That is, in general:
- $\mathbf a \times \paren {\mathbf b + \mathbf c} = \paren {\mathbf a \times \mathbf b} + \paren {\mathbf a \times \mathbf c}$
for $\mathbf a, \mathbf b, \mathbf c \in \R^3$.
Proof
We draw a triangular prism whose parallel edges are in the direction of $\mathbf a$ and with its end faces as triangles with sides $\mathbf b$, $\mathbf c$ and $\mathbf b + \mathbf c$.
From Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors, the vector areas of these triangular end faces are $\dfrac {\mathbf b \times \mathbf c} 2$ and $\dfrac {\mathbf c \times \mathbf b} 2$.
The remaining vector areas are $\mathbf b \times \mathbf a$, $\mathbf c \times \mathbf a$ and $\mathbf a \times \paren {\mathbf b + \mathbf c}$.
From Total Vector Area of Polyhedron is Zero:
- $\paren {\mathbf b \times \mathbf a} + \paren {\mathbf c \times \mathbf a} + \paren {\mathbf a \times \paren {\mathbf b + \mathbf c} } + \dfrac {\mathbf b \times \mathbf c} 2 + \dfrac {\mathbf c \times \mathbf b} 2 = 0$
from which we get:
- $\paren {-\mathbf b \times \mathbf a} + \paren {-\mathbf c \times \mathbf a} = \paren {\mathbf a \times \paren {\mathbf b + \mathbf c} }$
The result follows from Vector Cross Product is Anticommutative:
- $\mathbf a \times \mathbf b + \mathbf a \times \mathbf c = \mathbf a \times \paren {\mathbf b + \mathbf c}$
$\blacksquare$
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {II}$: The Products of Vectors: $6$. Application to Vector Products