Which product distributes over vector addition?

Which product distributes over vector addition?

• A. Cross Product
• B. Dot Product
• C. Tensor Product
• D. Vector Addition

Answer: (A)

Distributive behavior over addition comes from linearity. The cross product is linear in each argument, so for any vectors a, b, c we have a × (b + c) = a × b + a × c, and likewise (a + b) × c = a × c + b × c. This bilinear nature means adding vectors in one input simply adds the corresponding cross products. For a quick check, take a = (1,0,0), b = (0,1,0), c = (0,0,1). Then a × (b + c) = a × (0,1,1) = (0, -1, 1), while a × b = (0,0,1) and a × c = (0,-1,0); their sum is (0, -1, 1), matching perfectly. So the cross product distributes over vector addition.

Distributive behavior over addition comes from linearity. The cross product is linear in each argument, so for any vectors a, b, c we have a × (b + c) = a × b + a × c, and likewise (a + b) × c = a × c + b × c. This bilinear nature means adding vectors in one input simply adds the corresponding cross products. For a quick check, take a = (1,0,0), b = (0,1,0), c = (0,0,1). Then a × (b + c) = a × (0,1,1) = (0, -1, 1), while a × b = (0,0,1) and a × c = (0,-1,0); their sum is (0, -1, 1), matching perfectly. So the cross product distributes over vector addition.