Here's what I got to prove where '.' is dot. A.B=A.C Then B=C True or false? If true, prove it in general terms, if false, provide a counter-example. Ok, I just need some body to comment on my little proof here, and any guidelines to make it more thorough or whatnot. I know that the dot product is commutative, A.(B+C)=A.B +A.C, but not sure if it really needs to be in my proof or not. Proof ------ Say A.B=N and A.C=N (where N is a scalar number) so if N=N Then A.B=A.C If I cancel the A's, I get B=C. Is that a good way to approach that, or is there a better way of expressing it?
You know that (A^-1)(A)=1 or the identity. Then (A^-1)(A)(B)=(A^-1)(A)C) with this we can multiply both sides and get 1(B)=1(C) or B=C The raeson that (A^-1)(A)=1 is because (A^-1) is the inverse for A.
What if A is the zero vector? Then A.B=A.C no matter what B and C are. And even if A <> 0 if you break A, B and C down into components, I think you will find that you can come up with other situations where A.(B-C) must equal 0 even though you know nothing about the values of B and C individually. Try it.
No. In the first place, there is no "A^{-1}" when you are talking about dot product. There is, start with, no "identity" since A.I= A would not make sense. A is a vector and the dot product of two vectors is a number, not a vector. You are not really using either commutative or distributive laws: you are using cancellation which is exactly what you are asked about: Is is true that when A.B= A.C, B MUST equal C. You cannot use what you are asked to prove. Here is a hint. Choose two vectors at right angles. Call them A and B. Now choose a third vector at right angles to A. Call it C. What are A.B and A.C. Does that answer your question?
a.b=a.c a.(b-c)=0 Therefore a is perpendicular to b-c. This does not imply b=c. Example (3 space): a=(1,0,0) b=(x,u,v) c=(x,s,t) where x,u,v,s,t may assume any values.
I thought this problem was going to be easy, but I keep on getting confused each time I come back here. Let's see if I get this straight mathman. Let's say that A and B are perpendicular to each other. Now another vector, C, is perpendicular to A and B. So A.B=0 and A.C=0, but this doesn't imply that B and C HAVE to equal each other? And one more thing. Example (3 space): a=(1,0,0) b=(x,u,v) c=(x,s,t) Just some clarification. Does x for vecter b and c have to be the same number?
Take A= (1,0,0), B= (0,1,0), and C= (0,0,1). It can't get any simpler than that. You also say: "And one more thing. Example (3 space): a=(1,0,0) b=(x,u,v) c=(x,s,t) Just some clarification. Does x for vecter b and c have to be the same number?" I have absolutely no idea. Generally speaking we do NOT use the same letter to represent two different numbers, but what was the context?