Math: Vectors
posted by Anonymous .
What conditions must be satisfied by the vectors "u" and "v" for the following to be true?
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"?

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 = (UxVx)i + (UyVy)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 UV squared =
(UxVx)^2 + (UyVy)^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 
No I don't understand what you did.