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Math Help - finding a unit vector orthogonal to two other vectors U and V

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    finding a unit vector orthogonal to two other vectors U and V

    The exercise: find a unit vector orthogonal to both U= (1,1,0) and V = (0,1,1).
    The answer is below.


    finding a unit vector orthogonal to two other vectors U and V-screen-shot-2013-04-14-4.35.28-pm.png

    My question is why is it true that because (U dot X) = x_1 + x_3 and (V dot X) is equal to x_2 + x_3 that both U and V are orthogonal to X?
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    Re: finding a unit vector orthogonal to two other vectors U and V

    Quote Originally Posted by kingsolomonsgrave View Post
    The exercise: find a unit vector orthogonal to both U= (1,1,0) and V = (0,1,1).
    The answer is below.
    My question is why is it true that because (U dot X) = x_1 + x_3 and (V dot X) is equal to x_2 + x_3 that both U and V are orthogonal to X?
    Do you understand how to find the dot product of vectors?

    Do you understand why U\cdot V=1~?
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    Re: finding a unit vector orthogonal to two other vectors U and V

    yup, I know that (U dot V) is = 1 because U dot V = (u1 * v1 + u2 * v2 +...+uN * vN)

    I think it see the problem more clearly now: Is it that we create a vector x and dot multiply it by both vectors U and V and then pick values that will give the dot product between X and V and X and V = 0 for both cases?

    In this case x_1 =x_2 = -x_3 works to make both dot products zero so we can create X = (1, 1, -1).

    If this is the correct procedure, is there a general way of creating an orthogonal vector, or is it done by inspection?
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    Re: finding a unit vector orthogonal to two other vectors U and V

    Hello, kingsolomonsgrave!

    \text{Find a unit vector orthogonal to both }\vec u= \langle1,1,0\rangle\text{ and }\vec v = \langle0,1,1\rangle.

    Why isn't this basic theorem being used?


    A vector \vec n orthogonal to \vec u and \vec v is given by: . \vec u \times \vec v

    Hence: . \vec n \;=\;\begin{vmatrix} i&j&k \\ 1&1&0 \\ 0&1&1\end{vmatrix} \;=\;i(1-0) - j(1-0) + k(1-0) \;=\;i - j + k \;=\;\langle1,\text{-}1,1\rangle

    A unit vector is: . \vec U_1 \:=\:\left\langle \tfrac{1}{\sqrt{3}},\:\tfrac{-1}{\sqrt{3}},\:\tfrac{1}{\sqrt{3}}\right\rangle

    Another is its negative: . \vec U_2 \:=\:\left\langle \tfrac{-1}{\sqrt{3}},\:\tfrac{1}{\sqrt{3}},\:\tfrac{-1}{\sqrt{3}}\right\rangle
    Thanks from kingsolomonsgrave
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