How Do You Simplify This Vector Expression?

[izchief360]

I'm trying to simplify the following expression:

(4u + 3v) ⋅ (4u − 2v) − ll 3u − 4v ll²

And I'm unsure how to proceed.

 

[Stephen Tashi]

Assuming the dot represents the inner product, what formulas do you know that give the properties of the inner product?

 

[izchief360]

Is the inner product equivalent to the dot product? The only relevant formula I know is the that of the dot product, but I am unsure of how to apply order of operations when dealing with vectors.

 

[Hawkeye18]

Yes, the inner product is the dot product (in the case of real spaces). Dot product is a multiplication, so it has a higher order then addition/subtraction, and the same order as multiplication by a scalar.

 

[Stephen Tashi]

Some properties of the dot product are listed in the "Properties" section of the Wikipedia article http://en.wikipedia.org/wiki/Dot_product

 

[izchief360]

Thanks folks, I solved it. The process included taking the inside terms of the entire first term and dotting them with the entire second term as follows:
(4u + 3v) ⋅ (4u − 2v)
[4u ⋅ (4u − 2v)] + [3v ⋅ (4u − 2v)]
16u² - 8u⋅v + 12u⋅v - 6v²

and for the second part, ll 3u − 4v ll² is equivalent to (3u - 4v)⋅(3u - 4v), and it's the same process as above. Then, just combine like terms.