To find the angle between two vectors A and B, we can use the dot product formula. The dot product of two vectors A and B is given by:
A · B = |A| * |B| * cos(θ)
where |A| and |B| are the magnitudes of vectors A and B, respectively, and θ is the angle between the two vectors.
Given:
A = 41 + j
B = 61 + 3j
First, let's calculate the magnitudes of vectors A and B:
|A| = sqrt(41^2 + 1^2) = sqrt(1682 + 1) = sqrt(1683)
|B| = sqrt(61^2 + 3^2) = sqrt(3721 + 9) = sqrt(3730)
Next, let's find the dot product of A and B:
A · B = (41)(61) + (1)(3) = 2503
Now, substitute these values back into the dot product formula:
2503 = sqrt(1683) * sqrt(3730) * cos(θ)
Solving for θ:
cos(θ) = 2503 / (sqrt(1683) * sqrt(3730))
cos(θ) = 2503 / sqrt(6279390)
cos(θ) = 0.063254
Finally, find the angle θ by taking the arccosine of 0.063254:
θ = arccos(0.063254)
Using a calculator, find the arccosine of 0.063254 to get the angle between the two vectors A and B.
