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Math Help - vectors

  1. #1
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    vectors

    I have 3 vectors a = 2i - j + k, b = i -j +k, c= -i - 2j + 2k

    I have to find:

    1) the value of w when a + wb is orthogonal to c

    2) The value(s) of w that make c at 45 degrees to -i + wk

    The answers are:

    1) 4/3

    2) 1,7

    I have tried this and no matter what I do I cannot get these answers. Please help. Though to be honest I'm not sure where to start other than:

    x1x2 + y1y2 + z1z2 = 0

    Can anyone help?
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  2. #2
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    Quote Originally Posted by JQ2009 View Post
    I have 3 vectors a = 2i - j + k, b = i -j +k, c= -i - 2j + 2k

    I have to find:

    1) the value of w when a + wb is orthogonal to c

    2) The value(s) of w that make c at 45 degrees to -i + wk

    The answers are:

    1) 4/3

    2) 1,7

    I have tried this and no matter what I do I cannot get these answers. Please help. Though to be honest I'm not sure where to start other than:

    x1x2 + y1y2 + z1z2 = 0

    Can anyone help?
    Both of these tasks use the definition of the dot product:
    cos(a,b) = (a*b) / (|a||b|) where |a| and |b| are the lengths of the vectors and a*b is the dot product of a and b. Cos(a,b) means the angle between vectors a and b.

    2) We have vectors c= -i - 2j + 2k and u = -i + wk . The angle wanted here is 45, and cos (45) = 1/sqrt(2).
    Also,
    c*u = (-1)(-1) + -2*0 + 2*w = 1 + 2w
    |c| = sqrt(1 +4 +4) = sqrt(9) = 3
    |u] = sqrt(1 + w)

    Then we use the definition of the dot product cos(a,b) = (a*b) / (|a||b|)
    to get an equation where we can solve w:

    1/sqrt(2) = (1+2w) / 3*sqrt(1+w)

    And we'll get w = 1 or w = 7.

    I can't seem to get 4/3 ( I get -2/3, I might have calculated something wrong) in 1) , but the idea is similar: vectors u and v can only be orthogonal to each other (= the angle between them is 90) when the dot product u*v = cos(90) = 0
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