# Unit Vector Perpendicular to Given Vector

I understand there are questions on the Math Exchange already, but upon analyzing them I have still not figured out how to solve my own problem.

My problem is to find all vectors that are perpendicular to the vector $$(1, -2, 5)$$, have the y-components be equal to 3 times the x-components, and have a length of 5.

I know that the first step is to set the dot product of my vector and another vector equal to zero. And the result is $$i-2j+5k=0$$, correct?

After that my method falls apart as I am trying to find ALL vectors perpendicular to my vector. Help would be appreciated!

• Your first equation is correct. Any vector satisfying that will be perpendicular to your given vector. Now create equations for the other conditions and solve. – Don Thousand Oct 23 '19 at 21:54
• Despite the title you’ve given this question, the problem doesn’t want you to find a unit vector. – amd Oct 23 '19 at 22:31
• You do realize that there’s an infinite number of vectors perpendicular to $(1,-2,5)$, yes? It looks like you’ve picked out just a single solution to the first equation that you formulated. The probability is zero that it happened to be the right one. – amd Oct 23 '19 at 22:32
• @amd I believe my answer below is sufficient, thanks! – ThatOneNerdyBoy Oct 23 '19 at 22:36

Start with all vectors possible, let's designate them $$(x;y;z)$$. Now choose all vectors that are perpendicular to your vector $$0 = (1; -2; 5) \cdot (x;y;z) = x -2y + 5z$$. This gives you one equation. The other two are $$y = 3x$$ and $$5^2 = x^2 + y^2 + z^2$$. Solve the equations and you'll get the result.