How to prove X·X is non-negative and X·X = 0 implies X = 0

[newtomath]

If X is an N vector, is it 1) possible for X*X to be negative? 2) if x*x=0, what is X.

I am having trouble writing the proper proof. for 1 I stated that it is impossible for X*X to be negative bc if x is positive, X*X is positive and if x is negative, -X*-X is still positive.

for 2 I stated the properties of multiplication; in order for a product to = 0 one of the components must be 0.

Can someone advise on the proper method for writing these proofs?

Thanks

 

[Dick]

What is X.X in terms of the components of X?

 

[newtomath]

Not sure what youare asking, but X*X is the dot product of X. X is an arbitrary vector.

 

[Dick]

X is a vector, it has components. I.e. X=(x1,x2,...,xn). What's X.X in terms of the components, the little x's?

 

[lanedance]

I think Dick is asking you to examine how the dot product is caclulated in terms of the components of the vector.

Say X = (x1, x2, .. ,xn)^T

How do you calculate the dot product X*X in terms of the xi's?

 

[newtomath]

X= ( X1, X2, X3)
the dot product should be X1^2+X2^2 +X3^2. Squares can't be negative.


if X1^2+X2^2 +X3^2 =0 then X1 X2 and X3 must be zero

Is this all there is to it?

 

[lanedance]

looks good to me

 

[lanedance]

though you should generalise it to the n dimensional case rather than just 3