Given a vector $v = (2, 8)$, determine all vectors $w$ in form $(x , y)$ that are orthogonal to $v$. Vector $w$ must be orthonormal and therefore is a unit vector.

Knowing that the dot product of $\langle u,w\rangle = 0$, I attempted to find a vector $w$ but my problem is how do I find all possible orthonormal vectors $w$ that are orthogonal? My initial instinct tells me it involves finding a basis but I'm not sure.

  • $\begingroup$ Hint: In the plane, a family of orthogonal vectors relative to one given vector is a one-parameter family. $\endgroup$ Jan 15, 2020 at 1:21
  • $\begingroup$ $2x+8y=0\implies x=-4y$ $\endgroup$ Jan 15, 2020 at 1:23
  • $\begingroup$ are you asking for all vectors orthogonal to $v$, or only unit vectors, or both? $\endgroup$ Jan 15, 2020 at 1:29
  • $\begingroup$ All vectors orthogonal to v with unit length so therefore unit vectors only. $\endgroup$
    – fastlanes
    Jan 15, 2020 at 2:49

1 Answer 1


$(8,-2)$ and all its multiples are orthogonal to $(2,8)$.

Can you normalize $(8,-2)$; i.e., scale it so it's a unit vector (divide by the length)?

Another approach: the orthogonality condition is $2x+8y=0;$ that means $x=-4y$.

The unit vector condition is $x^2+y^2=1$.

Can you solve $(-4y)^2+y^2=1$ for $y,$ and then solve for $x$ too?


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