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.
$(8,-2)$ 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?