Question

Proof the cosine law of vector addation

Answer 1

The cosine law of vector addition, also known as the law of cosines, is a generalization of the Pythagorean theorem for non-right triangles. It's a formula that relates the magnitudes of three vectors and the cosine of the angles between them.

Given three vectors A, B, and C, where C = A + B, we can find the magnitude of the resulting vector C by applying the cosine law. The cosine law states that:

|C|^2 = |A|^2 + |B|^2 - 2 * |A| * |B| * cos(θ)

Where |A|, |B|, and |C| are the magnitudes of vectors A, B, and C, respectively, and θ is the angle between vectors A and B.

Proof:

1. Express vector C in terms of vectors A and B:

C = A + B

2. Compute the magnitude of vector C squared (|C|^2):

|C|^2 = (A + B) • (A + B)

Here • denotes the dot product of vectors.

3. Apply the properties of the dot product:

|C|^2 = A • A + A • B + B • A + B • B

4. Note that the dot product is commutative, so A • B = B • A:

|C|^2 = |A|^2 + 2 * A • B + |B|^2

5. Recall the geometric interpretation of the dot product: A • B = |A| * |B| * cos(θ), where θ is the angle between vectors A and B:

|C|^2 = |A|^2 + 2 * |A| * |B| * cos(θ) + |B|^2

6. Finally, rearrange the formula to match the cosine law:

|C|^2 = |A|^2 + |B|^2 - 2 * |A| * |B| * cos(θ)

Thus, the law of cosines (cosine law of vector addition) is derived and proven.