I will be using != for not equal
The cancellation law for real numbers is quite obvious: if a,b, and c != 0 are real numbers for which ac = bc then a = b. It is natural to ask if the cancellation law holds for three dimensional vectors.
If U x W = V x W, U (dot) W = V (dot) W, and W != 0, is it true that U = V? If Yes Prove it. If no give a counter example.
u•w = v•w
u•w - v•w = 0
(u-v)•w = 0
so u-v is orthogonal to w
u×w = v×w
u×w - v×w = 0
(u-v)×w = 0
so u-v is parallel to w
so, u-v = 0
u=v
To determine if the cancellation law holds for three-dimensional vectors, we need to analyze the given conditions and see if they imply that U = V, assuming W ≠ 0.
Let's break down the problem step by step:
1. The given conditions are:
a) U x W = V x W
b) U · W = V · W
c) W ≠ 0
2. First, let's consider the cross product condition:
U x W = V x W
The cross product of two vectors results in a vector that is orthogonal (perpendicular) to both of the original vectors. So, U x W and V x W are two vectors orthogonal to W.
3. Next, let's look at the dot product condition:
U · W = V · W
The dot product of two vectors yields a scalar value, representing the projection of one vector onto the other. The condition U · W = V · W implies that the projections of U and V onto W are equal.
Now, we need to determine if the given conditions (1 and 2) imply that U = V.
To prove or counterexample this statement, let's consider the situation with an example:
Suppose we have U = <1, 1, 0>, V = <2, 2, 0>, and W = <0, 0, 1>. Let's test if the conditions hold.
1. Cross product condition:
U x W = <1, 1, 0> x <0, 0, 1> = <1, -1, 0>
V x W = <2, 2, 0> x <0, 0, 1> = <2, -2, 0>
Since the cross products U x W and V x W are equal, the first condition is satisfied.
2. Dot product condition:
U · W = <1, 1, 0> · <0, 0, 1> = 0
V · W = <2, 2, 0> · <0, 0, 1> = 0
The dot products U · W and V · W are both zero, indicating that the projections of U and V onto W are equal.
Now, let's see if U = V:
U = <1, 1, 0>
V = <2, 2, 0>
Since U and V have different values despite satisfying the given conditions, this is a counterexample. Therefore, the cancellation law does not hold for three-dimensional vectors.
Please note that this counterexample does not imply that the cancellation law is always false for three-dimensional vectors. It merely shows that it does not hold in general.