Hi I have an HW question:
Given any vector $(x,y) in R$, provide a formula in terms of $x$ and $y$ for all unit vectors that are orthogonal to $(x, y)$
My answer is:
Let any unit vector be defined as $(a,b)$ which should satisfy the following equation
$ax+by=0$
for it to be orthogonal to $(x, y)$
That seemed a bit too easy which raises my doubt on whether i understood the question correctly or not. So did I? Did I answer is correctly?
Geometrically, if you have a vector $(x,y)\in \mathbb{R}^2$, what do vectors orthogonal to it look like? Can you find a formula for one such vector? (That is, can you think of one solution for $a$ and $b$ in your equation?)
If you can, then you have found one vector orthogonal to $(x,y)$. Now you presumably know how to turn that into a unit vector. Can you also figure out how to find the other unit vector that is orthogonal once you have the first one?
Now, you need to solve the equation you have got for $a$ and $b$. Since you have one eq. in two Variables, then you will have one free variable, say $b=t$. Subs back in the eq. To find $a$ and you are done. See my answer here.
