Is a perpendicular unit vector to another vector unique?

Question

Vector PQ = [-4, 1]. I have already worked out that a perpendicular vector is vector [1, 4] and the unit vector of that perpendicular vector is 1/√17[1, 4]. They were previous questions I had to answer. The last part of the question asks:

Is the answer unique? My answer was "The answer is one of an infinite amount of possible unit vectors that are perpendicular to PQ." I was told it is wrong.

Any ideas of what the correct answer might be?

Juan

Answer 1

You have an infinite number of unit vectors perpendicular to the given vector if you are working in $ \mathbb{R}^3$. However, in $\mathbb{R}^2$, you will only have two unit vectors perpendicular to the given vector. (both the vector you have calculated $1/\sqrt17[1,4]$ and the vector in the opposite direction $1/\sqrt17 [-1,-4]$)

Answer 2

Since your vector is a unit vector, it must have length $1$, so there are only 2 choices: $\vec{v}$ and $-\vec{v}$.