dot product

(noun)

A scalar product.

Related Terms

Examples of dot product in the following topics:

  • The Dot Product

    • The dot product takes two vectors of the same dimension and returns a single value.
    • The dot product takes two vectors and returns a single value.
    • The dot product can only be taken from two vectors of the same dimension.
    • The dot product is the sum of the product of the corresponding parameters.
    • Geometrically, the dot product is the product of the magnitudes of two vectors and the cosine of the angle between them.

  • Tangent Vectors and Normal Vectors

    • As we covered in another atom, one of the manipulations of vectors is called the Dot Product.
    • When you take the dot product of two vectors, your answer is in the form of a single value, not a vector.
    • In order for two vectors to be normal to each other, the dot product has to be zero.

  • The Cross Product

    • The cross product of two vectors is a vector which is perpendicular to both of the original vectors.
    • The cross product is a binary operation of two three-dimensional vectors.
    • If the two original vectors are parallel to each other, the cross product will be zero.
    • The cross product is denoted as $a \times b = c$.
    • The cross product is different from the dot product because the answer is in vector form in the same number of dimensions as the original two vectors, where the dot product is given in the form of a single quantity in one dimension.

  • 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.

  • Product Labeling

    • Labels serve to capture the attention of shoppers as well as provide useful information regarding the product.
    • The use of catchy words may cause strolling customers to stop and evaluate the product.
    • A label is a carrier of information about the product.
    • Examples of environmental and recycling symbols include the recycling symbol, the resin identification code, and the "green dot."
    • The Fair Packaging and Labeling Act (FPLA) is a law that applies to labels on many consumer products that states the products identity, the company that manufactures it, and the net quantity of contents.

  • DNA Analysis Using Genetic Probes and PCR

    • Example of genetic analysis method using PCR and immobilized oligonucleotide probes: The reverse dot-blot method has several unique properties that are valuable in a diagnostic setting: (1) The typing that results from a single sample can be located on a single strip.
    • This minimizes user labor as well as error potential and allows the use of standardized reagents. (3) Unlike dot-blot/oligonucleotide typing, only the PCR product is labeled, eliminating the potential problem of probes labeled to different specific activities.
    • In many PCR-based typing assays, the target DNA of interest is amplified and labeled by PCR, and the labeled products are hybridized to an array of immobilized diagnostic probes.
    • Spoligotyping, a reverse dot blot assay that detects the presence of a series of unique spacers in the direct repeat (DR) locus, meets the need for a simple and rapid method by which to distinguish M. tuberculosis complex strains.

  • 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.

  • Dot Plots

    • Judge whether a dot plot would be appropriate for a given data set
    • Dot plots can be used to display various types of information.
    • Each dot represents a single M & M.
    • A dot plot showing the number of M & M's of various colors in a bag of M & M's.
    • A dot plot showing the number of people playing various card games on a Wednesday.

  • Rhythmic Values

    • Dots and ties allow for basic durations to be lengthened.
    • A dot occurs after a pitch or a rest, and it increases its duration by half.
    • Generally, undotted notes divide into two notes; dotted notes divide into three.
    • Multiple dots can be added to a duration.
    • Subsequent dots add half the duration of the previous dot.