Math: Vectors

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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"?

  • Math: Vectors -

    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

  • Math: Vectors -

    No I don't understand what you did.

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