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Dot Product

The dot product can be defined by

{\bf X}\cdot {\bf Y} = \vert{\bf X}\vert\, \vert{\bf Y}\vert\cos \theta,
\end{displaymath} (1)

where $\theta$ is the angle between the vectors. It follows immediately that ${\bf X}\cdot {\bf Y} = 0$ if ${\bf X}$ is Perpendicular to ${\bf Y}$. The dot product is also called the Inner Product and written $\left\langle{a,b}\right\rangle{}$. By writing
$\displaystyle A_x$ $\textstyle =$ $\displaystyle A\cos \theta_A \qquad B_x=B\cos \theta_B$ (2)
$\displaystyle A_y$ $\textstyle =$ $\displaystyle A\sin \theta_A \qquad B_y=B\sin \theta_B,$ (3)

it follows that (1) yields
$\displaystyle {\bf A}\cdot {\bf B}$ $\textstyle =$ $\displaystyle AB\cos (\theta_A-\theta_B)$  
  $\textstyle =$ $\displaystyle AB(\cos \theta_A\cos \theta_B +\sin \theta_A\sin \theta_B)$  
  $\textstyle =$ $\displaystyle A\cos \theta_AB\cos \theta_B+A\sin \theta_AB\sin \theta_B$  
  $\textstyle =$ $\displaystyle A_xB_x+A_yB_y.$ (4)

So, in general,
{\bf X}\cdot {\bf Y} = x_1y_1+\ldots +x_ny_n.
\end{displaymath} (5)

The dot product is Commutative
{\bf X}\cdot {\bf Y} = {\bf Y}\cdot {\bf X},
\end{displaymath} (6)

(r{\bf X})\cdot {\bf Y} = r({\bf X}\cdot {\bf Y}),
\end{displaymath} (7)

and Distributive
{\bf X}\cdot ({\bf Y}+{\bf Z}) = {\bf X}\cdot {\bf Y}+{\bf X}\cdot {\bf Z}.
\end{displaymath} (8)

The Derivative of a dot product of Vectors is
{d\over dt} [{\bf r}_1(t)\cdot {\bf r}_2(t)]
= {\bf r}_1(t)...
...d{\bf r}_2\over dt} + {d{\bf r}_1\over dt} \cdot {\bf r}_2(t).
\end{displaymath} (9)

The dot product is invariant under rotations
$\displaystyle {\bf A}'\cdot{\bf B}'$ $\textstyle =$ $\displaystyle A_i' B_i'=a_{ij} A_j a_{ik} B_k = (a_{ij}a_{ik})A_jB_k$  
  $\textstyle =$ $\displaystyle \delta_{jk} A_jB_k = A_jB_j = {\bf A}\cdot{\bf B},$ (10)

where Einstein Summation has been used.

The dot product is also defined for Tensors $A$ and $B$ by

A\cdot B \equiv A^\alpha B_\alpha.
\end{displaymath} (11)

See also Cross Product, Inner Product, Outer Product, Wedge Product


Arfken, G. ``Scalar or Dot Product.'' §1.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 13-18, 1985.

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© 1996-9 Eric W. Weisstein