Dot Product Question: How to Solve (2a-5b)dot(b+3a) with Unit Vectors?

[Random-Hero-]

Homework Statement

I'm really at a loss here, if anyone could help me out I'd really appreciate it.


Given 'a' and 'b' unit vectors,

if |a+b| = root3, determine (2a-5b)dot(b+3a)

 

[Mark44]

|a + b|^2 = (a + b)\bullet(a + b)
and |a + b| = sqrt(3) ==> |a + b|^2 = 3

Now, use the fact that the dot product is associative, distributive, and commutative and the two equations above to see if you can evaluate (2a - 5b) )\bullet (b + 3a).

 

[Random-Hero-]

well I end up getting 13ab + 6a^2 - 5b^2

The answer is -11/2

I just can't seem to figure out how to get there :s

 

[Mark44]

Work with (a + b) \cdot (a + b) = 3. You also know that a and b are unit vectors, which means that a \cdot a = 1 and b \cdot b = 1.

 

[Random-Hero-]

Would I do like

(a+b)dot(a+b)=3

1 + 2ab + 1 = 3

ab = 1/2

then sub 1/2 into the ab and then get 6.5 + 6a^2 - 5b^2 and solve from there?

 

[Mark44]

Sort of, except that what you show as 6a^2 and -5b^2 is really 6a\cdot a and -5b\cdot b.