If A + B + C = 0. Show that A x B = B x C = C x A

Answer 1

To prove that A x B = B x C = C x A when A + B + C = 0, we can start by assuming that A, B, and C are non-zero vectors.

Using the cross product properties, we know that the cross product of two vectors is orthogonal to both vectors.

Thus, for A x B = B x C, the cross product of A and B is orthogonal to A and B individually.

Similarly, for B x C = C x A, the cross product of B and C is orthogonal to B and C individually.

Since A + B + C = 0, we can rewrite this equation as A = -B - C.

Now, we can substitute this value of A into the equations A x B = B x C and B x C = C x A:

-B - C x B = B x C (substituting A = -B - C)
-B x B - C x B = B x C (using the distributive property)
0 - C x B = B x C (since B x B = 0 due to A x A = 0 for any vector A)
-C x B = B x C

Similarly, we can substitute A = -B - C into the equation B x C = C x A:

B x C = C x (-B - C) (substituting A = -B - C)
B x C = -C x B - C x C (using the distributive property)
B x C = -C x B - (C x C) (since C x C = 0)
B x C = -C x B

Thus, we have proven that A x B = B x C = C x A when A + B + C = 0.