$\begingroup$

Let's say I have a vector:

$$ v = \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix}. $$

I was wondering how many vectors would be orthogonal to it. I noticed that these are some of the vectors orthogonal to it.$$ \begin{bmatrix}-1 \\ 0 \\ 1\end{bmatrix}, \begin{bmatrix}1 \\ 0 \\ -1\end{bmatrix}, \begin{bmatrix}1 \\ -1 \\ 0\end{bmatrix}, \begin{bmatrix}-1 \\ 1 \\ 0\end{bmatrix}, \begin{bmatrix}-2 \\ 1 \\ 1\end{bmatrix}. $$

Obviously, there are more. Is there a principled way to count the number of orthogonal vectors to a given vector? Thanks!

$\endgroup$ 4

3 Answers

$\begingroup$

Ummm... an infinite number, of course:

Find a single vector orthogonal to yours, and rotate it by an arbitrary angle around your vector.

And if you also allow vectors of different magnitude, well then...

$\endgroup$ 0 $\begingroup$

Notice that any vector of the form

$$\begin{pmatrix}x \\ y \\ -x-y\end{pmatrix}$$

will be orthogonal to

$$\begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix}$$

Thus, there is an infinite number of such vectors, more precisely a continuum vectors spanning a two-dimensional subspace.

$\endgroup$ $\begingroup$

Think of a plane orthogonal to a vector $(a,b,c)$ passing through $(x_0, y_0, z_0)$. This plane is given by the following:

$$ 𝑎(𝑥−𝑥0)+𝑏(𝑦−𝑦0)+𝑐(𝑧−𝑧0)=0 $$

For an intuition read this answer. Then, since we can have a plane orthogonal to a given vector, and there's an infinite number of vectors living on a plane, we must have an infinite number of vectors orthogonal to a given vector.

Moreover, since we also need to specify a point on which the plane passes to, there are also infinite orthogonal planes, each of which have infinite vectors orthogonal to that given vector.

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy