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

We will be using the concept of the dot product of two vectors for solving this.

Answer: The two unit vectors that make an angle of 60° with v = (6, 8) are (-0.39282, 0.919615 ) and (0.99282, -0.119615) respectively.

Let's understand this in more detail.

Explanation:

Given: Two unit vectors that make an angle of 60° with v = (6, 8)

Let a = (x, y) be a unit vector

where, |a| = x² + y² = 1

Applying dot product of unit vector with v, we get:

⇒ a.v = |a||v|cos Ø

⇒ a.v = 1. √(x² + y²) cos 60º

⇒ (x, y).(6, 8) = 1.√(6² + 8²) cos 60º

⇒ 6x + 8y = 1 × 10 × 1/2

⇒ 6x + 8y = 5

⇒ y = (5 - 6x)/8 -------(1)

Putting this value of y in modular equation of a, we get:

x² + y² = 1

⇒ x² + [(5 - 6x)/8]² = 1

⇒ 64x² + 25 - 60x + 36x² = 64

⇒ 100x² - 60x - 39 = 0

Solving the equation using quadratic formula, we get

x = [-b ±√(b² - 4ac)]/2a

x = [-(-60) ±√{(-60)² - 4×100×(-39)}] / 2 × 100

x = (3 ± 4√3)/10

x = -0.39282 , 0.99282

Putting this value of x in eq(1), we find y as:

y = (1/8)[5 - 6(-0.39282)] = 0.919615

y = (1/8)[5 - 6(0.99282)] = -0.119615

Thus, two unit vectors that make an angle of 60° with v = 6, 8 are (-0.39282, 0.919615 ) and (0.99282, -0.119615) respectively.