Final answer:
The statement that the sum and difference of two perpendicular vectors are equal is incorrect because the vectors point in different directions, and thus cannot be proved.
Step-by-step explanation:
The statement (a vector + b vector) = (a vector - b vector) is actually incorrect when vectors a and b are perpendicular to each other.
When subtracting vectors, we add the negative of the vector we're subtracting, that is A - B = A + (-B).
This applies to perpendicular vectors as well. If vectors a and b are perpendicular, their dot product is zero, meaning that they have no component in the direction of the other vector.
Thus, when we calculate a + b and a - b, we're actually getting two vectors that are perpendicular to each other, and not the same vector.
To illustrate, if a is represented by the x-axis and b by the y-axis, a + b would give us a vector going diagonally in the positive x and y direction, while a - b would result in a vector in the positive x direction and the negative y direction (essentially a reflection over the x-axis). These two will not be equal as they point in different directions.
Hence, the initial statement can't be proved because it's based on a false premise.