Find two unit vectors that make an angle of 60° with v = 3, 4

Solution:

Given, two unit vectors that make an angle of 60° with v = 3, 4.

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∅

a.v = 1.√( x² + y²) cos60° 

(x, y).(3,4) = 1.√( x² + y²) cos60°

3x + 4y = 1.√( 3²) + (4²) cos60°

3x + 4y = 1.√(9+16) (1/2)

3x + 4y = 1 × 5 × (1/2)

3x + 4y = 5/2

4y = 5/2 - 3x

y = (1/4)(2.5 - 3x) ------------ (1)

Put the value of y in modular equation,

x² + y² = 1

x²+((2.5 - 3x)/4)² = 1

16x² + 6.25 + 9x² - 15x = 16

25x² - 15x + 6.25 - 16 = 0

25x² - 15x - 9.75 = 0

Solving the equation using quadratic formula,

\(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\)

Here, a = 25, b = -15, c = -9.75

\(\\x=\frac{-(-15)\pm \sqrt{(-15)^{2}-4(25)(-9.75)}}{2(25)}\\x=\frac{15\pm \sqrt{225+975}}{50}\\x=\frac{15\pm \sqrt{1200}}{50}\\x=\frac{15\pm34.64}{50}\)

\(\\x=\frac{15+34.64}{50}\\x=\frac{49.64}{50}\\x=0.9928\)

\(\\x=\frac{15-34.64}{50}\\x=\frac{-19.64}{50}\\x=-0.3928\)

Putting the values of x,

y = (2.5 - 3(0.9928))/4 = -0.1196

y = (2.5 - 3(-0.3928))/4 = 0.9196

Therefore, the two unit vectors are (-0.3928, 0.9196) and (0.9928, -0.1196).

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Find two unit vectors that make an angle of 60° with v = 3, 4

Summary:

Two unit vectors that make an angle of 60° with v = 3, 4 are (-0.3928, 0.9196) and (0.9928, -0.1196).