## Friday, September 23, 2011

### 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.

References:

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