10.4: Matrix Formulation

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The dot product can also be written in matrix form. To begin, let's represent vectors as column vectors-that is, \(3 \times 1\) matrices. We'll define the vectors \(\mathbf{A}\) and \(\mathbf{B}\) as the column vectors

\[
\mathbf{A}=\left(\begin{array}{c}
A_{x} \\
A_{y} \\
A_{z}
\end{array}\right) ; \quad \mathbf{B}=\left(\begin{array}{c}
B_{x} \\
B_{y} \\
B_{z}
\end{array}\right)
\]

The dot product can then be written

\[
\mathbf{A} \cdot \mathbf{B}=\mathbf{A}^{T} \mathbf{B}=\left(\begin{array}{lll}
A_{x} & A_{y} & A_{z}
\end{array}\right)\left(\begin{array}{c}
B_{x} \\
B_{y} \\
B_{z}
\end{array}\right)=\left(A_{x} B_{x}+A_{y} B_{y}+A_{z} B_{z}\right) .
\]

This the the product of a \(1 \times 3\) row array with a \(3 \times 1\) column array, which gives a \(1 \times 1\) result (i.e. a scalar).

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