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!
Best Answer
A hint. The thing is, you need all prependicular vectors.
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.