If for two vectors A and B, sum (A+B) is perpendicular to the difference (Ā-B). The ratio of
their magnitude is
Answers
Explanation:
Well, this one is fun. A+B will, by force, result in a vector. Thus if u⃗ +v⃗ u→+v→ is <u1,u2>+<v1,v2><u1,u2>+<v1,v2> the result is <u1+v1,u2+v2><u1+v1,u2+v2> and is not a scalar. Same for subtraction, which is just adding the negative directions, so those two are not going to apply.
Multiplication is also a vector calculation and is seen by the formula for vectors AA and BB of: A×B=∥A∥∥B∥sinθnA×B=‖A‖‖B‖sinθn. And the magnitude of the vector is a formula you should have learned in the first few days of dealing with them for your study, but it’s basically an expanded Pythagorean calculation depending on how many dimensions you need to work through. nn, of course, is the unit vector, and sinθsinθ is the angle between, which for a perpendicular vector is…. well…. 1. So, here we have ∥A∥∥B∥n.‖A‖‖B‖n.
So, three formulas give us vectors, not a scalar. That leaves only one, the dot product, that can give us a simple number. By default it has to be right, and by fortune it is also correct, as the formula for the dot product of two vectors aa and bb is ∥a∥∥b∥cosθ‖a‖‖b‖cosθ, and that gives us the trick — the cosine of a perpendicular angle is zero, which wipes out the whole thing.