# Find two unit vectors that make an angle of 60° with v = 8, 6

**Solution:**

It is given that

Angle = 60°

v = 8, 6

Consider unit vector a = (x, y)

x^{2} + y^{2} = 1 …. (1)

Now apply the dot product of unit vector with v

a.v = |a||v|cos Ø

a.v = √ (x^{2} + y^{2}) √ (8^{2} + 6^{2}) cos 60º

8x + 6y = 1 × 10 × ½

So we get

8x + 6y = 5 …. (2)

y = (5 - 8x) / 6

Substituting the value of y in equation (1)

x^{2} + y^{2} = 1

x^{2} + [(5 - 8x)/6]^{2} = 1

x^{2} + 25/36 + 16x^{2}/9 - 20x/9 = 1

25x^{2}/9 - 20x/9 = 11/36

So we get

4 × (25x^{2} - 20x) = 11

100x^{2} - 80x = 11

100x^{2} - 80x - 11 = 0

Now solve the quadratic

x = [-b +-sqrt(b^{2} - 4ac)]/2a

x= { 2/5 ± √5/10}

Substituting the value of x in equation (2)

y = 3/10 ± 2√5/15

Therefore, the two unit vectors are (2/5 +√5/10, 3/10 + 2√5/15 ) and (2/5 -√5/10, 3/10 - 2√5/15).

## Find two unit vectors that make an angle of 60° with v = 8, 6

**Summary:**

The two unit vectors that make an angle of 60° with v = 8, 6 are (2/5 +√5/10, 3/10 + 2√5/15 ) and (2/5 -√5/10, 3/10 - 2√5/15).

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