Question

if vectors a+2b and 5a-4b are perpendicular to each other and a and b are unit vectors. Find the angle between a and b.

Answer 1

Let θ be the angle between a and b. We know that cosθ = (a∙b) / (|a||b|). Since a and b are unit vectors, |a|=|b|=1. Therefore we need to find a∙b.

a+2b and 5a-4b are perpendicular, so

(a+2b)∙(5a-4b)=0

Multiply this out to obtain

5a∙a + 6a∙b - 8b∙b = 0

which results in

6a∙b = 8b∙b - 5a∙a

Since a and b are unit vectors,

6a∙b = 8-5 = 3

Therefore, cos θ = 3/6. θ=π/3

Answer 2

Good!

Answer 3

Good

Answer 4

If two vectors are perpendicular their dot product will equal zero.
(a+2b)•(5a-4b)=0
5a•a+10a•b-4a•b-8b•b=0
6a•b=8b•b-5a•a
6a•b=8-5=3
cosθ=3/6.
θ=π/3