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calculus

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Find two unit vectors that make an angle of 60° with v = ‹3, 4›. Give your answers correct to three decimal places.

I'm not sure what to do here; I believe that I use to definition of a dot product (or maybe cross product) with relation to the trig function (cos for dot, sin for cross) but for whatever reason I can't get my answer to come out.

  • calculus -

    let such a vector be (a,b)
    then (a,b)⋅(3,4) = |(a,b)||(3,4)|cos60°
    3a + 4b = 5√(a^2 + b^2)(1/2)
    6a+8b = 5√(a^2+b^2)
    let a= 1 , any non-zero number will do
    6+8b = 5√(1+b^2)
    square both sides
    36+96b +64 b^2 = 25 + 25b^2
    39b^2 + 96b + 11 = 0
    b = (-96 ± √7500)/48
    = appr. -2.341 or -0.1205

    so two vectors could be (1, -2.341) or (1, -.1205)
    but you want a unit vector, so using the second point,
    |(1,-.1205)| = 1.00723

    giving us a unit vector of (1/1.00723 , -.1205/1.00723)
    or appr. (.9928 , -.1196)

    check:
    (3,4).(.9928,-.1196) = 2.5
    |(3,4)||(.9928,-.1196)cos60 = 2.5

    reverse the direction of the above vector to get
    (-.9928, .1196) for a second vector, or
    let a = some other value and repeat the above steps

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