Dot Product

The dot product can be defined by

$${\bf X}\cdot {\bf Y} = \vert{\bf X}\vert\, \vert{\bf Y}\vert\cos \theta,$$ (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

$$A_x = A\cos \theta_A \qquad B_x=B\cos \theta_B$$ (2)

$$A_y = A\sin \theta_A \qquad B_y=B\sin \theta_B,$$ (3)

it follows that (1) yields

$${\bf A}\cdot {\bf B} = AB\cos (\theta_A-\theta_B)$$

$$= AB(\cos \theta_A\cos \theta_B +\sin \theta_A\sin \theta_B)$$

$$= A\cos \theta_AB\cos \theta_B+A\sin \theta_AB\sin \theta_B$$

$$= A_xB_x+A_yB_y.$$ (4)

So, in general,

$${\bf X}\cdot {\bf Y} = x_1y_1+\ldots +x_ny_n.$$ (5)

The dot product is Commutative

$${\bf X}\cdot {\bf Y} = {\bf Y}\cdot {\bf X},$$ (6)

Associative

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

and Distributive

$${\bf X}\cdot ({\bf Y}+{\bf Z}) = {\bf X}\cdot {\bf Y}+{\bf X}\cdot {\bf Z}.$$ (8)

The Derivative of a dot product of Vectors is

$${d\over dt} [{\bf r}_1(t)\cdot {\bf r}_2(t)] = {\bf r}_1(t)... ...d{\bf r}_2\over dt} + {d{\bf r}_1\over dt} \cdot {\bf r}_2(t).$$ (9)

The dot product is invariant under rotations

$${\bf A}'\cdot{\bf B}' = A_i' B_i'=a_{ij} A_j a_{ik} B_k = (a_{ij}a_{ik})A_jB_k$$

$$= \delta_{jk} A_jB_k = A_jB_j = {\bf A}\cdot{\bf B},$$ (10)

where Einstein Summation has been used.

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

$$A\cdot B \equiv A^\alpha B_\alpha.$$ (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.