How to use the result of two orthogonal vectors to find two vectors that are orthogonal to $v=(2,-3)$

Question

If I have two vectors, for example: $v=(a,b)$ and $w=(-b,a)$ are orthogonal.

and then the questions is how to use the result to find two vectors that are orthogonal to $v=(2,-3) $?

imagine that $a$ and $b$ are both any number

Answer 1

Let $a=2$ and $b=-3$; what is $(-b,a)$?

Now let $-b=2$ and $a=-3$; what is $(a,b)$?

Answer 2

Recall that for two orthogonal vectors dot product is zero, thus we need that

$$(x,y)\cdot (2,-3)=2x-3y=0$$

and then to obtain two orthogonal vectors $(x_1,y_1)$ and $(x_2,y_2)$ let for example

• $x_1=3 \implies y_1=\frac23 x_1$

• $x_2=-3 \implies y_2=\frac23 x_1$

Answer 3

All the orthogonal vectors of $v=(a,b)$ are generated by the vector subspace $\langle(b,-a)\rangle$. So in your case the orthogonal vectors of $(2,-3)$ are all the multiple of $(-3,-2)$