dot product

(noun)

A scalar product.

Related Terms

Examples of dot product in the following topics:

  • More on vectors

    • Look again at the dot or inner product of two finite length vectors:
    • We can certainly use the same formula for the dot product of two infinite dimensional vectors:
    • It would be unconventional to use the "dot" for the inner product of functions, although we could.
    • The standard notation for the dot product of functions is $(f,g)$, thus
    • this means that the dot product of any $P_\ell$ with any other is zero.

  • Potentials and Charged Conductors

    • Given that work is the difference in final and initial potential energies (∆U), we can relate this difference to the dot product of force at every infinitesimal distance l along the path between the points within the conductor:
    • Rewriting U as the product of charge (q) and potential difference (V), and force as the product of charge and electric field (E), we can assert:

  • Angular Momentum Transport

    • $\displaystyle \dot L^+ = \dot M \left ( GM r \right )^{1/2}.$
    • $\displaystyle \dot L^- = \beta \dot M \left ( GM r_I \right )^{1/2}$
    • $\displaystyle \tau = f_\phi \left ( 2 \pi r \right ) \left ( 2 h \right ) ( r ) = \dot L^+ - \dot L^- = \dot M \left [ \left ( GM r \right )^{1/2} - \beta \left ( GM r_I \right )^{1/2} \right ]$
    • The viscous torque is the product of the viscous stress in the tangential direction, the area upon which the stress acts (the half-height of the disk is $h$) and the radius.
    • $\displaystyle \eta = \frac{\dot M}{6 \pi r^2 h \Omega } \left [ \left ( GM r \right )^{1/2} - \beta \left ( GM r_I \right )^{1/2} \right ].$

  • Some Special Matrices

    • $I_n = \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & \dots \\ 0 & 1 & 0 & 0 & \dots \\ 0 & 0 & 1 & 0 & \dots \\ \vdots & & & \ddots & \\ 0 & \dots &0 & 0 & 1 \end{array} \right].$
    • $\mbox{diag}(x_1, x_2, \cdots , x_n) = \left[ \begin{array}{ccccc} x_1 & 0 & 0 & 0 & \dots \\ 0 & x_2 & 0 & 0 & \dots \\ 0 & 0 & x_3 & 0 & \dots \\ \vdots & & & \ddots & \\ 0 & \dots &0 & 0 & x_n \end{array} \right].$
    • Another interpretation of the matrix-vector inner product is as a mapping from one vector space to another.
    • To see this first notice that for any matrix $A$ , the inner product $(A\cdot \mathbf{x}) \cdot \mathbf{y}$ , which we write as $(A\mathbf{x},\mathbf{y})$ , is equal to $(\mathbf{x},A^T\mathbf{y})$ , as you can readily verify.
    • Now, as you already know, and we will discuss shortly, the inner product of a vector with itself is related to the length, or norm, of that vector.

  • Non-relativistic particles

    • Let's assume that $|\beta| \ll 1$ and focus on a particular frequency $\nu$ so that ${\dot u} \sim u \nu$.
    • $\displaystyle \frac{E_{acc}}{E_{vel}} \sim \frac{R \dot{u}}{c^2} \sim \frac{R u \nu}{c^2} = \frac{u}{c} \frac{R}{\lambda}$
    • Let use the angle $\Theta$ to denote the angle between the vectors ${\bf n}$ and $\dot{{\bf u}}$, so we have
    • $\displaystyle |{\bf E}_{acc}| = |{\bf B}_{acc}| = \frac{q \dot{u}}{Rc^2} \sin \Theta$
    • $\displaystyle P = \frac{d W}{d t} = \frac{q^2 \dot{u}^2}{4\pi c^3} \int \sin^2\Theta d\Omega = \frac{q^2 \dot{u}^2}{2 c^3}\int_{-1}^1 \left (1 - \mu^2 \right ) d \mu \\ \displaystyle = \frac{2 q^2 \dot{u}^2}{3c^3}$

  • Relation Between Electric Potential and Field

    • They share a common factor of inverse Coulombs (C-1), while force and energy only differ by a factor of distance (energy is the product of force times distance).
    • ∆P can be substituted for its definition as the product of charge (q) and the differential of potential (dV).
    • We can then replace W with its definition as the product of q, electric field (E), and the differential of distance in the x direction (dx):
    • The presence of an electric field around the static point charge (large red dot) creates a potential difference, causing the test charge (small red dot) to experience a force and move.

  • Radiation Reaction

    • $\displaystyle -{\bf F}_{rad} \cdot {\bf u} = \frac{2 q^2 \dot{u}^2}{3 c^3}$
    • $\displaystyle -\int_{t_1}^{t_2} {\bf F}_{rad} \cdot {\bf u} dt = \frac{2 q^2 }{3 c^3} \int_{t_1}^{t_2} \dot{\bf u} \cdot \dot{\bf u} dt = \frac{2 q^2 }{3 c^3} \left [ \left .
    • \dot{\bf u} \cdot {\bf u} \right |_{t_1}^{t_2} - \int_{t_1}^{t_2} \ddot{\bf u} \cdot \dot{\bf u} dt \right ].$

  • Simple harmonic oscillation

    • Then $x(0) = 0$ and $\dot{x}(0) = \omega_0 B$.
    • taking $t=0$ as the time at which the spring passes through the origin with velocity $\dot{x}(0)$.
    • It's a little cleaner if we absorb the product of $\omega_0$ and $ t_0$ as a single, dimensionless phase constant $\Delta$.
    • where $a = A \cos(\Delta)$ and $b = -A \sin(\Delta)$$x(0), \dot{x}(0); A,t_0; A,\Delta; a,b$.
    • $E = T + U = \frac{1}{2} m \dot{x}^2 + \frac{1}{2} k x^2$

  • Basic Map Types

    • Dot maps use dots to show comparative densities of features over a base map.
    • The dots are all the same size, and each dot represents a certain unit of the feature of interest.
    • For instance, if you wanted to display the relative densities of cattle across the continental US, if one city had 2,223 cattle and you decided to represent 500 cattle with one dot, that city would have a cluster of four dots.
    • As do dot maps, graduated symbol maps use symbols that occur at points across a map.

  • How about relativistic particles?

    • Let's start by assuming the $\beta$ is parallel to ${\dot{\beta}}$, so $\beta \times {\dot{\beta}}=0$.
    • $\displaystyle P = 2\pi \frac{q^2 {\dot u}^2}{4\pi c^2} \int_{-1}^1 \frac{1-\mu^2}{\left ( 1 - \beta \mu \right )^5} d\mu = \frac{2}{3} \frac{q^2 {\dot u}^2}{c^2} \gamma^6$
    • $\displaystyle \frac{dP(t')}{d\Omega} \approx \frac{q^2 {\dot u}^2}{4\pi c^3} \frac{ \theta^2 }{\left [ 1 - (1 - \gamma^{-2}/2 ) (1 - \theta^2/2 ) \right ]^5} \\ \displaystyle \approx \frac{q^2 {\dot u}^2}{4\pi c^3} \frac{ \theta^2 }{\left [ \gamma^{-2}/2 + \theta^2/2 \right ]^5} \\ \displaystyle \approx \frac{8}{\pi} \frac{q^2 {\dot u}^2}{c^3} \gamma^8 \frac{ (\gamma \theta)^2 }{\left [ 1 + (\gamma \theta)^2 \right ]^5}$
    • Let's repeat the calculation for circular motion, in which $\beta \perp {\dot{\beta}}$.
    • For circular motions the applied force is $\gamma m {\dot u}$, yielding