Final answer:
The angle between two nonzero vectors can be found by calculating the scalar product of their components, and then using the inverse cosine of that product divided by the product of their magnitudes to get the angle.
Step-by-step explanation:
To find the angle between two nonzero vectors, you should first identify the individual components of each vector along your chosen x- and y-axes. If your vectors are represented in two dimensions, they will have 'x' and 'y' components; for three dimensions, they will have 'z' components as well. Once you've determined these components (Ax, Ay, Bx, By, and potentially Az, Bz), you can use them to calculate the scalar (dot) product of the two vectors.
The formula for the scalar product is A · B = Ax × Bx + Ay × By + Az × Bz. According to the properties of the dot product, when vectors are unit vectors, their product will be zero if they are perpendicular and one if they are parallel. With the calculated dot product and the magnitudes of the vectors (denoted as A and B), you can find the cosine of the angle between the vectors using the following equation: cos(φ) = (A · B) / (A × B).
To get the actual angle, take the inverse cosine of the result. This will give you the angle φ, which is measured from vector A to vector B and ranges from 0° to 180°. The use of inverse cosine ensures that the angle is always measured in the smallest possible positive angle.