Question

If 'a' and 'b' are unit vectors that make an angle of 60 degrees with each other, calculate
l 3a - 5b l and l 8a + 3b l

*the 'a' and 'b' have a carat of top of them*

How do i answer this without using components?

Answer 1

From point O draw the two unit vectors a and b with a 60° angle between them.
Extend vector a to a point A, so that OA = 3 units
In the opposite direction of vector b, draw OB = 5 units
Complete the parallelogram AOBC.
You now have vector OC = 3a - 5b , where OB = 5, BC = 3, angle B = 60°

cosine law:
|OC|^2 = 5^2 + 3^2 - 2(3)(5)cos60°
= 19
|3a - 5b| = √19

I will leave it up to you to find the 2nd one in a similar way

Answer 2

check using components
let a = (1,0)
then b = (1/2, √3/2)
3a - 5b = (3,0) - (5/2 , 5√3/2)
|3a - 5b| = √((5/2 - 3)^2 + (5√3/2-0)^2 )
= √( 1/4 + 75/4) = √19