To distinguish points and vector’s matrix, there’s a homogeneous representation that we use a 4x1 matrix and the last row serves as a “type” of the matrix
3D point: $\begin{bmatrix} 2\\ 5\\ 7\\ 1 \end{bmatrix}$ 3D vector $\; \begin{bmatrix} 2\\ 5\\ 7\\ 0 \end{bmatrix}$
Doing so allow us to maintain the properties such as “point - vector → point”, “point - point → vector”.
If an affine combination is performed on such format, the type information is also preserved as well.
(recall that an affine combination is that $P = a_1P_1 + a_2P_2$ where $a_1 + a_2 = 1$ and a convex is a special case where $a_1,a_2\ge0$)
To bring back the homogeneous representation to the normal form, we can just remove the last row
Vectors can make up a basis, also known as a frame.