Cross Product

Inner product, not cross product. Look up the axioms for what a function has to satisfy to be an inner product

 

For your final answer you'll actually need to prove either all three parts of the definition are satisfied, or demonstrate that one of them isn't for each one

For example

ac-bd is not an inner product since <(1,1), (1,1)> = 0 even though it's of the 'form' ac+bd with constants

 

You have the standard inner product stuck in your head. It's true that in Euclidean-space (the usual space we're used to working with), the inner product is defined as:
[tex]<(x_1,x_2),(y_1,y_2)>=x_1y_1+x_2y_2[/tex]
but this is just one possible definition of the inner product. It turns out that you can define lots of different inner products (in terms of the vector's components) as long as you follow three rules:
1) Symmetry: [itex]<\mathbf{x},\mathbf{y}>=<\mathbf{y},\mathbf{x}> [/itex]
2) Linearity: [itex]<a\mathbf{x}+b\mathbf{y},\mathbf{z}>=a<\mathbf{x},\mathbf{z}>+b<\mathbf{y},\mathbf{z}> [/itex]
3) Positive-Definiteness: [itex]<\mathbf{x},\mathbf{x}>\geq 0, \mathbf{x}\neq 0[/itex]

You need to check to see if those different inner products satisfy these conditions.

 

You didn't actually show that symmetry fails, you just said it did. You have to find two vectors u and v, and show that <u,v> is not equal to <v,u>

 

You still haven't shown anything. How do you know 4ad+7bc=/=4cb+7da? Pick two ACTUAL vectors, with ACTUAL numbers. It doesn't need to be that complicated.

And I demonstrated how something of the form (first*first) + (last*last) can not be an inner product: ac-bd. Then the inner product of (1,1) with itself comes out to

1*1-1*1 = 1-1=0. So while your intuition was correct, you need a more formal proof because the way you described was, in its entirety, incorrect

 

There you go! Now that looks like a compelling reason why that is not a well-defined inner product.