Yes, the dot product is zero.

To form a 2D column matrix that is perpendicular to an other:

Swap the elements, and negate one of them.

This only works in 2D, however.
It gives you one of *an infinite number* columns orthogonal
to the given one.
For example, all the following vectors are orthogonal to ( -5, 3)^{T}:

- ( 3, 5 )
^{T} - ( -3, -5)
^{T} - ( 1.5, 2.5)
^{T} - ( 6, 10)
^{T} - .... and so on

The reason this works is:
If **u** is orthogonal to **v**, then **u · v** = 0.
So (k**u**) **· v** = k(**u · v**) = 0,
for any real number k.
So there are an infinite number of vectors (k**u**) orthogonal to **v**.

Often one wishes to find a *unit normal* to a given vector.
A **unit normal** to a given vector is a vector that:

- Is orthogonal (normal) to the given vector.
- Has a length of one.

Remember not to confuse the two ideas *normalizing a vector*
(making a unit vector in the same direction as the vector),
and *computing a unit normal* (making a unit vector in an orthogonal
direction to a vector.)