Dot/Scalar/Inner Product of Vectors in Arbitrary Non Standard Basis

1. The Dot/Scalar/Inner Product of 2 Vectors \(A\) and \(B\), both given in terms of Same Arbitraty Set of 2 Basis Vectors \(e_1\) and \(e_2\) is calculated as follows
   
   \(A= A_1e_1 + A_2e_2\hspace{5mm}B= B_1e_1 + B_2e_2\)
   
   \(A \cdot B= (A_1e_1 + A_2e_2).(B_1e_1 + B_2e_2)=A_1B_1(e_1.e_1) + A_1B_2(e_1.e_2) + A_2B_1(e_2.e_1 ) + A_2B_2(e_2.e_2) \)
   
   \(\Rightarrow A \cdot B= \begin{bmatrix}A_1 & A_2\end{bmatrix} \begin{bmatrix}e_1\\e_2\end{bmatrix} \begin{bmatrix}e_1 & e_2\end{bmatrix}\begin{bmatrix}B_1 \\ B_2\end{bmatrix}= B \cdot A=\begin{bmatrix}B_1 & B_2\end{bmatrix} \begin{bmatrix}e_1 \\ e_2\end{bmatrix} \begin{bmatrix}e_1 & e_2\end{bmatrix} \begin{bmatrix}A_1 \\ A_2\end{bmatrix}\)
   
   \(\Rightarrow A \cdot B= \begin{bmatrix}A_1 & A_2\end{bmatrix} \begin{bmatrix}e_1.e_1 & e_1.e_2\\e_2.e_1 & e_2.e_2\end{bmatrix}\begin{bmatrix}B_1 \\ B_2\end{bmatrix}= B \cdot A=\begin{bmatrix}B_1 & B_2\end{bmatrix} \begin{bmatrix}e_1.e_1 & e_1.e_2\\e_2.e_1 & e_2.e_2\end{bmatrix}\begin{bmatrix}A_1 \\ A_2\end{bmatrix}\)  (...1)
   
   Similarly, the Dot/Scalar/Inner Product of 2 Vectors \(A\) and \(B\), both given in terms of Same Arbitraty Set of 3 Basis Vectors \(e_1\), \(e_2\) and \(e_3\) is calculated as follows
   
   \(A= A_1e_1 + A_2e_2 + A_3e_3\hspace{5mm}B= B_1e_1 + B_2e_2 + B_3e_3\)
   
   \(A \cdot B= (A_1e_1 + A_2e_2 + A_3e_3).(B_1e_1 + B_2e_2 + B_3e_3)\)
   
   \(\Rightarrow A \cdot B= A_1B_1(e_1.e_1) + A_1B_2(e_1.e_2) + A_1B_3(e_1.e_3) + A_2B_1(e_2.e_1 ) + A_2B_2(e_2.e_2) + A_2B_3(e_2.e_3) + A_3B_1(e_3.e_1)+ A_3B_2(e_3.e_2) + A_3B_3(e_3.e_3) \)
   
   \(\Rightarrow A \cdot B= \begin{bmatrix}A_1 & A_2 & A_3\end{bmatrix} \begin{bmatrix}e_1\\e_2\\e_3\end{bmatrix} \begin{bmatrix}e_1&e_2&e_3\end{bmatrix} \begin{bmatrix}B_1 \\ B_2 \\ B_3\end{bmatrix}= B \cdot A=\begin{bmatrix}B_1 & B_2 & B_3\end{bmatrix} \begin{bmatrix}e_1\\e_2\\e_3\end{bmatrix} \begin{bmatrix}e_1& e_2& e_3\end{bmatrix}\begin{bmatrix}A_1 \\ A_2 \\ A_3\end{bmatrix}\)
   
   \(\Rightarrow A \cdot B= \begin{bmatrix}A_1 & A_2 & A_3\end{bmatrix} \begin{bmatrix}e_1.e_1 & e_1.e_2 & e_1.e_3\\e_2.e_1 & e_2.e_2 & e_2.e_3\\e_3.e_1 & e_3.e_2 & e_3.e_3\end{bmatrix}\begin{bmatrix}B_1 \\ B_2 \\ B_3\end{bmatrix}= B \cdot A=\begin{bmatrix}B_1 & B_2 & B_3\end{bmatrix} \begin{bmatrix}e_1.e_1 & e_1.e_2 & e_1.e_3\\e_2.e_1 & e_2.e_2 & e_2.e_3\\e_3.e_1 & e_3.e_2 & e_3.e_3\end{bmatrix}\begin{bmatrix}A_1 \\ A_2 \\ A_3\end{bmatrix}\)  (...2)
   
   Hence, the Dot/Scalar/Inner Product of 2 Vectors \(A\) and \(B\), both given in terms of Same Arbitraty Set of \(N\) Basis Vectors \(e_1\), \(e_2\), \(e_3\),..., \(e_n\) is calculated as follows
   
   \(A= A_1e_1 + A_2e_2 + ... + A_ne_n\hspace{.5cm}B= B_1e_1 + B_2e_2 + ... + B_ne_n\)
   
   \(A \cdot B= \begin{bmatrix}A_1 & A_2 & \cdots & A_n\end{bmatrix}\begin{bmatrix}e_1\\e_2\\ \vdots \\ e_n\end{bmatrix} \begin{bmatrix}e_1&e_2& \cdots & e_n\end{bmatrix}\begin{bmatrix}B_1 \\ B_2 \\\vdots\\ B_n\end{bmatrix}= B \cdot A=\begin{bmatrix}B_1 & B_2 & \cdots & B_n\end{bmatrix}\begin{bmatrix}e_1\\e_2\\ \vdots \\ e_n\end{bmatrix} \begin{bmatrix}e_1&e_2& \cdots & e_n\end{bmatrix}\begin{bmatrix}A_1 \\ A_2 \\\vdots\\ A_n\end{bmatrix}\)
   
   \(\Rightarrow A \cdot B= \begin{bmatrix}A_1 & A_2 & \cdots & A_n\end{bmatrix}\begin{bmatrix}e_1.e_1 & e_1.e_2 & \cdots & e_1.e_n\\e_2.e_1 & e_2.e_2 & \cdots & e_2.e_n\\ \vdots & \vdots & \ddots & \vdots \\ e_n.e_1 & e_n.e_2 & \cdots & e_n.e_n\end{bmatrix}\begin{bmatrix}B_1 \\ B_2 \\\vdots\\ B_n\end{bmatrix}= B \cdot A=\begin{bmatrix}B_1 & B_2 & \cdots & B_n\end{bmatrix}\begin{bmatrix}e_1.e_1 & e_1.e_2 & \cdots & e_1.e_n\\e_2.e_1 & e_2.e_2 & \cdots & e_2.e_n\\ \vdots & \vdots & \ddots & \vdots \\ e_n.e_1 & e_n.e_2 & \cdots & e_n.e_n\end{bmatrix}\begin{bmatrix}A_1 \\ A_2 \\\vdots\\ A_n\end{bmatrix}\)  (...3)
   
   
2. Dot/Scalar/Inner Product of 2 Vectors given in terms of Different Sets of Basis Vectors can be calculated Only if all the Basis Vectors have Same Dimension. The following calculates the Dot/Scalar/Inner Product of Vector \(A\) given in terms of Basis Vector Set \(e_1, e_2, ..., e_n\) and Vector \(B\) given in Basis Vector Set \(e_1', e_2', ..., e_m'\)
   
   \(A= A_1e_1 + A_2e_2 + ... + A_ne_n\hspace{.5cm}B= B_1e_1' + B_2e_2' + ... + B_me_m'\)
   
   \(A \cdot B= \begin{bmatrix}A_1 & A_2 & \cdots & A_n\end{bmatrix}\begin{bmatrix}e_1\\e_2\\ \vdots \\ e_n\end{bmatrix}\begin{bmatrix}e_1'&e_2'& \cdots & e_m'\end{bmatrix}\begin{bmatrix}B_1 \\ B_2 \\\vdots\\ B_m\end{bmatrix}= B \cdot A=\begin{bmatrix}B_1 & B_2 & \cdots & B_m\end{bmatrix}\begin{bmatrix}e_1'\\e_2'\\ \vdots \\ e_m'\end{bmatrix}\begin{bmatrix}e_1&e_2& \cdots & e_n\end{bmatrix}\begin{bmatrix}A_1 \\ A_2 \\\vdots\\ A_n\end{bmatrix}\)
   
   \(A \cdot B= \begin{bmatrix}A_1 & A_2 & \cdots & A_n\end{bmatrix}\begin{bmatrix}e_1.e_1' & e_1.e_2' & \cdots & e_1.e_m'\\e_2.e_1' & e_2.e_2' & \cdots & e_2.e_m'\\ \vdots & \vdots & \ddots & \vdots \\ e_n.e_1' & e_n.e_2' & \cdots & e_n.e_m'\end{bmatrix}\begin{bmatrix}B_1 \\ B_2 \\\vdots\\ B_m\end{bmatrix}= B \cdot A=\begin{bmatrix}B_1 & B_2 & \cdots & B_m\end{bmatrix}\begin{bmatrix}e_1'.e_1 & e_1'.e_2 & \cdots & e_1'.e_n\\e_2'.e_1 & e_2'.e_2 & \cdots & e_2'.e_n\\ \vdots & \vdots & \ddots & \vdots \\ e_m'.e_1 & e_m'.e_2 & \cdots & e_m'.e_n\end{bmatrix}\begin{bmatrix}A_1 \\ A_2 \\\vdots\\ A_n\end{bmatrix}\)  (...4)
   
   
3. As can be seen in calculations given equations (1), (2), (3) and (4) above, the Dot/Scalar/Inner Product between 2 Vectors can be represented in form of Matrix Multiplication Consisting of the Product of 3 Matrices of following types
   1. The First Matrix is a Row Matrix / Co-Vector which consists of Components of 1 of the 2 Vectors.
   2. The Third Matrix is a Column Matrix / Vector which consists of Components of the other Vector.
   3. The Second Matrix or Middle Matrix is formed by Multiplying the Transpose of the Matrix of Basis Vector Set of the First Vector with the Matrix of Basis Vector Set of the Second Vector. If Basis Vector Sets for both Vectors are Same, this Matrix is known as the Gram Matrix / Gramian Matrix / Metric Tensor. If Basis Vector Sets for both Vectors are Different, this Matrix is known as the Mixed Metric Tensor.
4. When Both Vectors are given in terms of Same Basis Vector Set, Pre-Multiplying a Vector with the Gram Matrix / Gramian Matrix / Metric Tensor or Post-Multiplying the Gram Matrix / Gramian Matrix / Metric Tensor to a Co-Vector, converts the Components of Vector/Co-vector to it's corresponding Covariant Components. Hence, calculating Dot/Scalar/Inner Product of 2 Vectors given in terms of Same Basis Vector Set is same as calculating the Dot Product between Contravariant Components of One Vector and Covariant Components of the Other Vector.
5. You can use the Dot/Scalar/Inner Product of Vectors in Arbitrary Basis Calculator to calculate Dot/Scalar/Inner Product of Vectors in Arbitrary Basis.