Chapter 3.7, Problem 18P

With the cross product of two vectors defined by (4.14), show that finding the cross product is a linear operation, that is, show that (4.18) is valid. Warning hint: Don’t try to prove it by writing out components: Writing, for example, $i{A}_x\times (j{B}_y+k{B}_x)=i{A}_x\times j{B}_y+i{A}_x\times k{B}_z$ would be assuming what you're trying to prove. Further hints: First show that (4.18) is valid if B and C are both perpendicular to A by sketching (in the plane perpendicular to A) the vectors B, C, B + C, and their vector products with A. Then do the general case by first showing that $A\times B$ and $A\times B_{\perp }$ (where $B_{\perp }$ is the vector component of B perpendicular to A) have the same magnitude and the same direction.