Given a and b unit vectors

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

First I foiled the whole thing:
=2(a dot b)+ 6 (a dot a) - 5(b dpt b) -15(a dot b)
=-13(a dot b) + 6(a dot a) - 5(b dot b)
=-13(1)(1)+6-5
=-13+1
=-12
Textbook answer is -11/2

I don't know how to solve this problem! How do I get |a+b| and what do I do with the square root of 3?

To determine the magnitude of the vector sum |a+b|, you need to consider the dot product between a and b. The dot product of two unit vectors is equal to the cosine of the angle between them.

1. Start by finding the dot product of a and b: a · b.
- If a and b are orthogonal (perpendicular), their dot product will be zero.
- If a and b are parallel (pointing in the same or opposite directions), their dot product will be 1 or -1.

2. Given that |a+b| = √3, the magnitude of the vector sum, you can square both sides of the equation to get rid of the square root:
(|a+b|)² = ( √3 )²
|a+b|² = 3

3. Express the magnitude of the vector sum in terms of the dot product:
|a+b|² = a · a + 2(a · b) + b · b

4. Substitute the dot product values into the equation:
3 = 1 + 2(a · b) + 1
3 = 2 + 2(a · b)

5. Simplify the equation:
2(a · b) = 3 - 2
2(a · b) = 1

6. Divide both sides by 2:
a · b = 1/2

Now that you know the value of a · b, you can use it to find the value of (2a-5b) · (b+3a).

Given (2a-5b) · (b+3a), you can apply the distributive property and simplify as follows:

(2a-5b) · (b+3a) = (2a-5b) · b + (2a-5b) · (3a)

1. Find the dot product of (2a-5b) · b:
(2a-5b) · b = 2(a · b) - 5(b · b)

2. Replace a · b with 1/2 (from the previous result) and simplify:
2(a · b) - 5(b · b) = 2(1/2) - 5(1)
= 1 - 5
= -4

3. Find the dot product of (2a-5b) · (3a):
(2a-5b) · (3a) = 2(a · a) - 5(b · a)

4. Substitute the value of a · a (which is 1) into the equation and simplify:
2(a · a) - 5(b · a) = 2 - 5(b · a)

5. Multiply -5 by 1/2 (since a · b is 1/2 from earlier) and simplify:
2 - 5(b · a) = 2 - 5(1/2)
= 2 - 5/2
= 4/2 - 5/2
= -1/2

Now, you can add the results of step 2 and step 5 to find the final answer:

(2a-5b) · (b+3a) = (2a-5b) · b + (2a-5b) · (3a)
= -4 + (-1/2)
= -8/2 - 1/2
= -9/2

Therefore, the value of (2a-5b) · (b+3a) is -9/2, which matches the textbook answer of -11/2.