They are both still examples of matrix multiplication. The first example is a 1x3 matrix multiplied by a 3x1 matrix, and the result is then a 1x1 matrix. The second example is a 2x1 matrix multiplied by a 1x2 matrix, and the result is a 2x2 matrix.

With "matrix multiplication". Think of matrix multiplication of a "dot product" of row and column. To find the number in "column i, row j" of the result, take the dot product of "row j" in the first matrix with "column i" in the second.

Specifically,
[tex]\begin{bmatrix}5 & 2 & 1\end{bmatrix}\begin{bmatrix}2 \\ -4 \\ 7\end{bmatrix}= 14[/itex]
The dot product of the single row in the first matrix with the single column in the second. A "row matrix" times a column matrix is a number.

[tex]\begin{bmatrix}2 \\ 1\end{bmatrix}\begin{bmatrix} -1 & 3\end{bmatrix}= \begin{bmatrix}-2 & 6 \\ -1 & 3\end{bmatrix}[/itex].
"Row 1 is just the number 2 and "column 1" is just the number -1. Their product is -2. "Column 2" is just the number 3 so the product of "row 1 and column 2" is (2)(3)= 6. "Row 2" is just 1. The product of "row 2 and column 1" is (1)(-1)= -1. The product of "row 2" and "column 2" is (1)(3).

For multiplication, you need everything to be in a multiplication space (my terminology) … both the input and the output.

The product of two matrices is always another matrix (possibly a different size and shape).

The dot product of two vectors isn't a vector, so the dot product isn't multiplication unless you regard everything (both vectors and scalars) as matrices.

So think of this as matrix multiplication, and don't use the phrase "vector multiplication".