If A x B = A x C, and A, B, C are nonzero vectors, does it follow that B =C ? Explain.
No. AxB = |A| * |B| * sinθ
AxC = |A| * |C| * sinØ
So, as long as |B| sinθ = |C| sinØ and A,B,C are co-planar, things will be equal.
If A,B and A,C form parallelograms of equal area, AxB = AxC
To answer this question, we need to understand the properties of vector multiplication and the conditions required for two vectors to be equal.
In general, vector multiplication follows the commutative property, meaning that for any vectors A and B, A x B = B x A. However, this property alone does not imply that A x B = A x C results in B = C.
To determine whether B and C are equal in this case, we can divide both sides of the equation A x B = A x C by A (assuming A ≠ 0, which is given). This gives us:
(B/A) x A = (C/A) x A
Next, we can use the cross product property that (K · A) x A = 0 for any scalar K and any vector A. Applying this property to both sides of the equation, we have:
(B/A) x A - (C/A) x A = 0
Factoring out A, we get:
[(B - C)/A] x A = 0
For the cross product of two vectors to be zero, either one or both of the vectors must be zero. Since A ≠ 0 (given that A is a nonzero vector), it follows that:
(B - C)/A = 0
Therefore, B - C = 0. In other words, B = C.
In conclusion, from the given statement A x B = A x C, where A, B, and C are nonzero vectors, it follows that B = C.