The dot product can be defined by
![\begin{displaymath}
{\bf X}\cdot {\bf Y} = \vert{\bf X}\vert\, \vert{\bf Y}\vert\cos \theta,
\end{displaymath}](d2_1481.gif) |
(1) |
where
is the angle between the vectors. It follows immediately that
if
is Perpendicular to
. The dot product is also called the Inner Product and written
.
By writing
it follows that (1) yields
So, in general,
![\begin{displaymath}
{\bf X}\cdot {\bf Y} = x_1y_1+\ldots +x_ny_n.
\end{displaymath}](d2_1495.gif) |
(5) |
The dot product is Commutative
![\begin{displaymath}
{\bf X}\cdot {\bf Y} = {\bf Y}\cdot {\bf X},
\end{displaymath}](d2_1496.gif) |
(6) |
Associative
![\begin{displaymath}
(r{\bf X})\cdot {\bf Y} = r({\bf X}\cdot {\bf Y}),
\end{displaymath}](d2_1497.gif) |
(7) |
and Distributive
![\begin{displaymath}
{\bf X}\cdot ({\bf Y}+{\bf Z}) = {\bf X}\cdot {\bf Y}+{\bf X}\cdot {\bf Z}.
\end{displaymath}](d2_1498.gif) |
(8) |
The Derivative of a dot product of Vectors is
![\begin{displaymath}
{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}](d2_1499.gif) |
(9) |
The dot product is invariant under rotations
where Einstein Summation has been used.
The dot product is also defined for Tensors
and
by
![\begin{displaymath}
A\cdot B \equiv A^\alpha B_\alpha.
\end{displaymath}](d2_1503.gif) |
(11) |
See also Cross Product, Inner Product, Outer Product, Wedge Product
References
Arfken, G. ``Scalar or Dot Product.'' §1.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL:
Academic Press, pp. 13-18, 1985.
© 1996-9 Eric W. Weisstein
1999-05-24