# Determine all orthonormal vectors (x, y) that are orthogonal to the vector v (2, 8).

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.

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

$$(8,-2)$$ 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?