Explanation:

The vector sum of two forces is perpendicular to their vector differences. This statement is based on the concept of vector addition and subtraction.

Vector Addition:
When two forces act on an object, their vector sum is obtained by adding the individual force vectors. The magnitude and direction of the resultant vector depend on the magnitudes and directions of the individual force vectors.

Vector Subtraction:
The vector difference between two forces is obtained by subtracting one force vector from the other. The magnitude and direction of the resultant vector depend on the magnitudes and directions of the individual force vectors.

Perpendicularity:
The given statement states that the vector sum of two forces is perpendicular to their vector difference. This means that the resultant vector is perpendicular to the direction of the force difference vector.

Proof:

Let's consider two forces, F1 and F2.

The vector sum of these forces is given by:
R = F1 + F2

The vector difference between these forces is given by:
D = F1 - F2

To find the dot product of R and D, we can write:
R · D = (F1 + F2) · (F1 - F2)

Expanding the dot product, we get:
R · D = F1 · F1 - F1 · F2 + F2 · F1 - F2 · F2

Since the dot product of two vectors is commutative, we can rewrite it as:
R · D = F1 · F1 - 2(F1 · F2) + F2 · F2

Now, let's consider the magnitudes of the forces:
|F1 · F1| = F1^2 (Magnitude of a vector squared is the dot product of the vector with itself)
|F2 · F2| = F2^2

Since the dot product of two perpendicular vectors is zero, we have:
R · D = F1^2 - 2(F1 · F2) + F2^2 = F1^2 + F2^2 - 2(F1 · F2) = 0

This equation implies that:
F1^2 + F2^2 = 2(F1 · F2)

If the magnitudes of the forces are equal, i.e., F1 = F2, then the equation becomes:
2F1^2 = 2(F1 · F1)
F1^2 = F1^2

This implies that the magnitudes of the forces are equal.

Therefore, the correct answer is option 'B': The forces are equal in magnitude.