Is length/distance a vector?

Question

I have heard that area is a vector quantity in 3 dimensions, e.g. this Phys.SE post, what about the length/distance? Since area is the product of two lengths, does this mean that length is also a vector quantity, and why?

Answer 1

Length and distance are not vector quantities (they are scalar quantities), but position and displacement are vector quantities (at least according to common terminological conventions). Here is how all of these are defined. Note that I am restricting the discussion here to vectors in three-dimensional Euclidean space $\mathbb R^3$.

Every point in three-dimensional space can be specified by a triple of real numbers $\mathbf x = (x,y,z)$ given its coordinates with respect to three axes. This triple is called the position of the point and is clearly a vector.

The length of any vector, such as a position vector, is defined as $$ |\mathbf x| = \sqrt{x^2 + y^2 + z^2} $$ Notice that, by definition, length is a positive real number. Given two points $\mathbf x_1=(x_1, y_1, z_1)$ and $\mathbf x_2=(x_2, y_2, z_2)$, the displacement vector pointing from point $1$ to point $2$ is defined as $$ \mathbf x_{21} = \mathbf x_2 - \mathbf x_1 = (x_2-x_1, y_2-y_1, z_2-z_1) $$ The length of the displacement vector is called the distance between the two points and is therefore given by $$ d(\mathbf x_1, \mathbf x_2) = |\mathbf x_{21}| = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} $$ Note. I have heard some using the terms distance and displacement interchangeably, or using the term displacement for what I have called distance, and using distance to refer to the total length of a path along which an object travels.

Having said all of this, there is in fact a product that allows one to construct area vectors given two position vectors. It's called the cross product. If you take the cross product $$ \mathbf x_1\times\mathbf x_2 $$ of two position vectors, then you get a vector whose length is the area of the parallelogram spanned by these vectors, and whose direction is perpendicular to this parallelogram.