Dot product problem

Question

a*b=|a||b|cosθ; |a|=1, |b|=4, θ=π/3

Find:

1) (2a-3b)(a+b)

2) |2a-3b|

How to solve these kind of problems?

Answer 1

1) (2a - 3b)(a + b) = 2a² + 2ab - 3ab - 3b² = 2IaI² - IaI·IbIcosθ - 3IbI² = 2·1² - 1·4cosπ/3 - 3·4² = - 48

2) I2a - 3bI² = (2a - 3b)² = 4IaI² - 12IaI·IbIcosθ + 9IbI² = 4·1² - 12·1·4·cosπ/3 + 9·16 =124;

I2a - 3bI = 124^1/2 = 2·31^1/2 = 2√31