What conditions must be satisfied by the vectors "u" and "v" for the following to be true?

Question

a) |u + v| = |u - v|
vector "u" is perpendicular to vector "v"

b) |u + v| > |u - v|
0° ≤ θ ≤ 90°

c) |u + v| < |u - v|
90° < θ ≤ 180°

------- Can you please explain to me why these conditions are true? Why is it perpendicular for the first one? Why is less less than 90°, but greater than 0° for question "b"? Why is it less than 180°, but greater than 90° for question "c"?

Answer 1

Look at resultants
I will call them L and R for left and right

a)
Slope of U = Uy/Ux
Slope of V = Vy/Vx = -1/slope of U if perpendicular = -Ux/Uy
so
Vy/Vx = - Ux/Uy
- Vx Ux = VyUy
Now U + V = (Ux+Vx)i + (Uy+Vy)j
and U - V = (Ux-Vx)i + (Uy-Vy)j
magnitude of U+V squared =
(Ux+Vx)^2 + (Uy+Vy)^2
= Ux^2 + 2 UxVx +Vx^2 +Uy^2+2 UyVy^2+Vy^2
magnitude of U-V squared =
(Ux-Vx)^2 + (Uy-Vy)^2
= Ux^2 [[[[[-2 UxVx ]]]] +Vx^2 etc.
SEE WHAT IS HAPPENING?
If UxVx= - UyVy
those middle terms disappear and the magnitudes squared are the same.
which means that + or - the square roots are the same which means the absolute values are the same

Answer 2

No I don't understand what you did.