# How to find orthogonal vectors?

by meteorologist1
Tags: orthogonal, vectors
 P: 290 How to find orthogonal vectors? The only condition on the vector is that its dot product with x vanishes? Just solve $$\vec{x}\cdot\vec{y} = 0$$ --J
 P: 610 since someone has already posted the answer... I will show you another alternative method for this problem let's say you have vector x in R^n, and wanna find a vector y orthogonal to it we know the dot product of these two vector must be zero $$\vec{x} \cdot \vec{y} = 0$$ $$\Rightarrow x_{1} y_{1} + x_{2} y_{2} + x_{3} y_{3} + .....+ x_{n-1} y_{n-1} + x_{n} y_{n} = 0$$ $$\Rightarrow x_{1} y_{1}+ x_{2} y_{2} + x_{3} y_{3} + .....+ x_{n-1} y_{n-1} = - x_{n} y_{n}$$ $$\Rightarrow y_{n} = -(x_{1} y_{1} + x_{2} y_{2} + x_{3} y_{3} +.....+ x_{n-1} y_{n-1}) / x_{n}$$ now, you have more freedom to vary $y_{1}.y_{2}.....y_{n-1}$ , as long as you the $y_{n}$ follows the formulas above