Math - Vectors - Dot/scalar/inner product

The dot product, scalar, or inner product is the vector multiplication which executes for two vectors \( \mathbf{A} \) and \( \mathbf{B} \) in three dimensions is [1, 2]

\[ \begin{align} \mathbf{A} \cdot \mathbf{B} &= \left(A_1 \mathbf{e}_1 +  A_2 \mathbf{e}_2 + A_3 \mathbf{e}_3 \right) \cdot \left(B_1 \mathbf{e}_1 +  B_2 \mathbf{e}_2 + B_3 \mathbf{e}_3 \right) \\ &= A_1 B_1 + A_2 B_2 + A_3 B_3 \end{align} \]

That is, the dot product multiplies each corresponding component of the vectors and adds them together to  obtain a scalar.  So you could also call this a vector component product.

In progress...to be continued.

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References:

[1] K. Karamcheti. Principles of Ideal-Fluid Aerodynamics. John Wiley & Sons, Inc., New York, NY. 1966

[2] W. Kaplan. Advanced Calculus (5th ed.). Addison-Wesley. 2002