Find two unit vectors that make an angle of 60° with v = ‹8, 6›. Give your answers correct to three decimal places.

**Answer 1**

Let’s call the unit vector . Since it’s a unit vector

a² + b² = 1

The cosine of the angle between and <8,6> is

cos(60°) = (8a + 6b)/√(8²+6²) = (1/10)(8a + 6b)

This is equal to ½. So there are two equations in the two unknowns a and b.

(1) 8a + 6b = 5

(2) a² + b² = 1

Isolating a in equation (1) and squaring

8a = 5 – 6b

64a² = 36b² – 60b + 25

Use a² = 1 – b²

64 – 64b² = 36b² – 60b + 25 ==> 100b² – 60b – 39 = 0

The quadratic formula gives two solutions

b = (3 ± 4√3)/10.

The corresponding a values are

a =(4 – 3√3)/10 and a =(4 + 3√3)/10, respectively.

So the vectors are

<(4 - 3√3)/10, (3 + 4√3)/10> and

<(4 + 3√3)/10, (3 - 4√3)/10>