How to find orthogonal vectors?

  1. Hi, this might be very easy, but I forgot how to do the following: I have a vector in R^6: (x1, x2, x3, x4, x5, x6). How do I find a vector such that their dot product vanishes? I know how to do it for the two dimensional case: (x1, x2), so the vector that is perpendicular to it is c(-x2, x1) where c is a scalar. Could someone show me how to do it for the vector in R^6 case?
  2. jcsd
  3. BobG

    BobG 2,364
    Science Advisor
    Homework Helper

    Take the cross product of your vector and any other vector that has the same origin (any unit vector usually fits the bill and is easiest since only one axis is non-zero).
  4. Ok I see. I'm not sure how to do a cross product for a six-component vector though. Could you please show me?
  5. The only condition on the vector is that its dot product with x vanishes?

    Just solve

    [tex]\vec{x}\cdot\vec{y} = 0[/tex]

  6. robphy

    robphy 4,416
    Science Advisor
    Homework Helper
    Gold Member

    Given a vector in n-dimensions, there is a whole (n-1)-dimensional hyperplane of vectors that are orthogonal to that vector.

    Let me do the 4D-case. The 6D-case is easily obtained by generalizing.
    Given V=(x1,x2,x3,x4), choose W=(w1,w2,w3,w4) such that
    0=(V dot W)=x1w1+x2w2+x3w3+x4w4.

    One choice is w1=-x2, w2=x1, w3=0, w4=0. (This is a generalization of your 2D example).

    Basically, you can choose values for three components of W. The fourth is determined by the requirement that 0=(V dot W).
  7. since someone has already posted the answer... I will show you another alternative method for this problem

    let's say you have vector x in R^n, and wanna find a vector y orthogonal to it
    we know the dot product of these two vector must be zero

    [tex] \vec{x} \cdot \vec{y} = 0 [/tex]

    [tex] \Rightarrow x_{1} y_{1} + x_{2} y_{2} + x_{3} y_{3} + .....+ x_{n-1} y_{n-1} + x_{n} y_{n} = 0 [/tex]

    [tex]\Rightarrow x_{1} y_{1}+ x_{2} y_{2} + x_{3} y_{3} + .....+ x_{n-1} y_{n-1} = - x_{n} y_{n} [/tex]

    [tex]\Rightarrow y_{n} = -(x_{1} y_{1} + x_{2} y_{2} + x_{3} y_{3} +.....+ x_{n-1} y_{n-1}) / x_{n} [/tex]

    now, you have more freedom to vary [itex] y_{1}.y_{2}.....y_{n-1} [/itex] , as long as you the [itex] y_{n} [/itex] follows the formulas above :biggrin:
  8. HallsofIvy

    HallsofIvy 40,678
    Staff Emeritus
    Science Advisor

    If you know how to do it with 2 dim vectors, just take a vector with all except two components 0 and fix the other 2 to give a 0 dot product!

    For example, to find a vector orthogonal to <1, -3, 4, 1 , -1, 2>, take all except the first two components 0, take the first two 3 and 1 respectively: <3, 1, 0, 0, 0, 0>.

    Of course, there are many independent vectors orthogonal to a given six dimensional vector (not true in 2 dimensions). You could also choose to make all except the first and last components 0: <2, 0, 0, 0, 0, -1>.

    OR <0, 4, 3, 0, 0, 0>

    OR ....
  9. All right thanks all.
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