Is a 1D vector the same as a unit vector?

Question

Is a 1D vector the same as a unit vector? For example, is A = 3 j a 1D vector or a unit vector? It's magnitude is greater than 1 so it should be a 1D vector, and NOT a unit vector.

Answer 1

A unit vector is a vector with magnitude $1$. So $3\hat j$ is certainly not a unit vector, since it has magnitude $3$.

The most common definition of a one-dimensional vector is an element of a one-dimensional vector space (if we instead define a one-dimensional vector as a scalar multiple of a unit vector then every vector is a one-dimensional vector, and the definition is useless). If we follow the convention that unit vectors are labelled $\hat i, \hat j, \hat k$ etc. then $3 \hat j$ is presumably a member of a vector space with at least two dimensions, so it is not a one-dimensional vector either. On the other hand, $3 \hat i$ could be a one-dimensional vector, as long as the vector space to which it belongs only has one dimension.

So there are vectors that are both unit vectors and one-dimensional vectors ($\hat i$ and $-\hat i$ in a one-dimensional vector space); vectors that are one-dimensional vectors but not unit vectors; vectors that are unit vectors but not one-dimensional vectors; and vectors like $3 \hat j$ which are neither.

Answer 2

You have answered your own question, the difference between a 1D vector and a unit vector is that unit vector has a magnitude of 1 where as a 1D vector is some scalar times the unit vector. Every unit vector is a 1D vector but not vice versa