To find two unit vectors that make an angle of 60° with the vector v = (3, 4), we will use the concept of the dot product and unit vectors.
First, let’s calculate the magnitude of vector v:
Magnitude of v = √(3² + 4²) = √(9 + 16) = √25 = 5.
Next, we find the unit vector in the direction of vector v. The unit vector u is given by:
u = (1/|v|) * v = (1/5) * (3, 4) = (3/5, 4/5).
Now, we want to find unit vectors that make an angle of 60° with v. The cosine of the angle θ between two vectors a and b can be computed as:
cos(θ) = (a · b) / (|a| |b|).
Since we are looking for unit vectors, their magnitudes will be 1, so the equation simplifies to:
cos(60°) = (u · w) / 1, where w is the desired unit vector.
Given that cos(60°) = 1/2, we have:
u · w = 1/2.
If we denote w as (x, y), then:
(3/5 * x + 4/5 * y) = 1/2.
Multiplying through by 5 gives us:
3x + 4y = 2.
Now, since w is a unit vector, it also satisfies:
x² + y² = 1.
Now, we can solve these two equations:
1. 3x + 4y = 2.
2. x² + y² = 1.
From the first equation, we can express y in terms of x:
y = (2 – 3x) / 4.
Substituting this into the second equation:
x² + ((2 – 3x) / 4)² = 1.
Clearing the fraction:
x² + (4 – 12x + 9x²) / 16 = 1.
Multiplying through by 16 gives:
16x² + 4 – 12x + 9x² = 16.
This simplifies to:
25x² – 12x – 12 = 0.
Now we can use the quadratic formula to find x:
x = [12 ± √(144 + 1200)] / 50.
x = [12 ± √(1344)] / 50.
x = [12 ± 36.66] / 50.
Calculating both possibilities:
1. x = 48.66 / 50 = 0.9732.
2. x = -24.66 / 50 = -0.4932.
Now plug these x-values back to find corresponding y-values:
For x ≈ 0.9732:
y = (2 – 3 * 0.9732) / 4 ≈ -0.235.
For x = -0.4932:
y = (2 + 3 * 0.4932) / 4 ≈ 0.578.
So the two unit vectors we have found are approximately:
1. (0.9732, -0.235).
2. (-0.4932, 0.578).
In conclusion, the two unit vectors that make an angle of 60° with the given vector (3, 4) are:
1. (0.9732, -0.235)
2. (-0.4932, 0.578).