Help with simple dot product proof

  • Thread starter Juntao
  • Start date
  • #1
45
0
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?
 

Answers and Replies

  • #2
Arden1528
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.
 
  • #3
1,036
1
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.
 
  • #4
HallsofIvy
Science Advisor
Homework Helper
41,833
956
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?
 
  • #5
mathman
Science Advisor
7,858
446
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.
 
Last edited:
  • #6
45
0
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?
 
  • #7
HallsofIvy
Science Advisor
Homework Helper
41,833
956
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?
 

Related Threads on Help with simple dot product proof

Replies
20
Views
2K
  • Last Post
Replies
8
Views
739
Replies
10
Views
3K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
3
Views
588
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
1K
Replies
2
Views
10K
Top