As z1 = 3 - 4i and z2 = -4 + 3i, then the acute angle between the vectors will be?
- 18°16'
- 17°16'
- 16°16'
- None of these
Explanation
To find the acute angle between the vectors, first find the dot product:
z1 = 3 - 4i
z2 = -4 + 3i
Dot product = (3)(-4) + (-4)(3) = -12 - 12 = -24
Next, find the magnitudes:
|z1| = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5
|z2| = √((-4)^2 + 3^2) = √(16 + 9) = √25 = 5
Now, find the cosine of the angle:
cos(θ) = (Dot product) / (|z1| |z2|)
= -24 / (5 * 5)
= -24 / 25
θ = arccos(-24/25) ≈ 16°16'
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