Find vector vertical to given vector of R^4

Question

Given the vector

a=(5,6,7,13)

find all the vectors vertical to it.

Let's say that the vector

c=(x,y,z,w)

is the vector we want to find. Then,

a c = 0

<=> 5x + 6y + 7z + 13w = 0

What I can do in addition to this to give the answer? Could you help, please?

Answer 1

By "vertical", I think you mean orthogonal.

You have obtained the answer already, you just have to write it in set notation:

$\{(x,y,z,w)^T\in\mathbb{R}^4\mid5x + 6y + 7z + 13w = 0\}$

If you want a particular vector, you can set arbitrary values for $x,y,z$ and then solve for $w$.

For instance, you can choose $x=0,y=0,z=1$. This forces $7+13w=0$, so $w=-7/13$.

So a particular orthogonal vector is $(0,0,1,-7/13)$.

Answer 2

You can express $x=-\dfrac{6}{5}y-\dfrac{7}{5}z-\dfrac{13}{5}w.$ Denote $y=5\alpha$, $z=5\beta$, $w=5\gamma$ and your solutions are: $$\mathbf{c}=(-6\alpha-7\beta-13\gamma,\;5\alpha,\;5\beta,\;5\gamma)\quad \alpha,\beta,\gamma\in\mathbb{R}.$$