Basic questions on matrix multiplication

Matrix operations, especially with vectors, are a bit confusing in Mathematica.

ClearAll[v1, v2, a, b, c, x, y, z];
v1 = {a, b, c}; m = Partition[Range[9], 3]; v2 = {x, y, z};

Use MatrixForm to display the expressions nicely. They are still just lists.

Map[MatrixForm, {v1, m, v2}];

Then the inner product or Dot gives a scalar as Bill said (1x3, 3x3, 3x1)

v1.m.v2

    (a + 4 b + 7 c) x + (2 a + 5 b + 8 c) y + (3 a + 6 b + 9 c) z

What you might mean is a "3BY1 1BY3" matrix multiplication

z1 = KroneckerProduct[v1, v2]

    {{a x, a y, a z}, {b x, b y, b z}, {c x, c y, c z}}

This next operation (with *) is not really a normal kind of matrix or tensor operation, so check your mathematics and your meaning, but you can do it.

z2 = z1 * m

    {{a x, 2 a y, 3 a z}, {4 b x, 5 b y, 6 b z}, {7 c x, 8 c y, 9 c z}}

Gives what you seek. Realize there are three kinds of operation used here: Dot/Inner, KroneckerProduct (see also TensorProduct and Outer), and Times.

Hope this helps.

(All--I'll take any kind of guidance or link in the chat room on how to better format answers and show 2D output in StackExchange.)