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Homework Statement
For what values of k is (scalar product of vectors a and b) = a_{1}b_{1}-a_{1}b_{2}-a_{2}b_{1}+ka_{2}b_{2} a valid scalar product?
The vectors a and b are defined as:
a = a_{1}e_{1} + a_{2}e_{2}
b = b_{1}e_{1} + b_{2}e_{2}
where e_{1} and e_{2} are unit vectors. a and b are also entirely real.
Homework Equations
Well, I know several properties of scalar products that must be true so that seems like the best way to start.
1) The scalar product of a vector (say a or b) with itself must have a real, numerical value corresponding to the dot product of the vector with itself.
2) The scalar product must be linear in each of its two members.
I also know that if the e vectors truly are unit, they will have magnitude of 1 when scalar producted (is that a word?) with themselves. They should also have a scalar product of magnitude 0 when scalar producted with each other if they are orthogonal. However, I have no idea if these unit vectors are necessarily orthonormal by definition or just normal. My reasoning that they aren't necessarily orthonormal is that the vectors that compose a basis of a Hilbert space, much like a set of unit vectors, are not necessarily orthogonal or even normal.
Finally, I know from the Schwarz Inequality that the magnitude of (the scalar product of a and b)^{2} \leq(scalar product of a and a)*(scalar product of b and b). However, I am not sure if that helps in this problem.
The Attempt at a Solution
Originally I looked toward property #2 above to solve this problem; that is, the vectors that we are taking a scalar product of need to be linearly independent. However, I cannot see of a way to prove that k must have a specific value based on this fact.
I also found another property in the book that I omitted from the previous section because it is so formatting heavy. For any two vectors (say a and b), if the components of each vector are real then (scalar product of a and b) can be written as:
(scalar product of a and b) = a_{1}b_{1} * (scalar product of e_{1} and e_{1}) + a_{1}b_{2} * (scalar product of e_{1} and e_{2}) + a_{2}b_{1} * (scalar product of e_{2} and e_{1}) + a_{2}b_{2} * (scalar product of e_{2} and e_{2})
I think it must have something to do with this property, as it so closely mimics the form of the problem. So I know that, assuming the unit vectors are normal, their scalar products with themselves are equal to 1. So, matching coefficients, k would just have to also be 1. But this seems far too simple and it fails to explain why the cross scalar products are negative. In fact, the negative, real value of the cross terms seems to tell me that the "unit" vectors are definitely not orthogonal. But now I feel as if I've talked myself in a circle again on what should be a relatively easy problem.
If anyone has any insight on this problem I'd really appreciate it. I've been beating my head against it for quite some time now and it's been bothering me quite a bit. If anyone wishes to know, the problem is from the Arfken Math Methods text. Thanks in advance.
Also, sorry for the shoddy formatting on the scalar products, I could not seem to get the syntax correct for them.