vector

Examples of vector in the following topics:

• Components of a Vector

  • All vectors have a length, called the magnitude, which represents some quality of interest so that the vector may be compared to another vector.
  • Vectors, being arrows, also have a direction.
  • To visualize the process of decomposing a vector into its components, begin by drawing the vector from the origin of a set of coordinates.
  • This is the horizontal component of the vector.
  • He also uses a demonstration to show the importance of vectors and vector addition.

• Adding and Subtracting Vectors Graphically

  • Draw a new vector from the origin to the head of the last vector.
  • Since vectors are graphical visualizations, addition and subtraction of vectors can be done graphically.
  • This new line is the vector result of adding those vectors together.
  • Then, to subtract a vector, proceed as if adding the opposite of that vector.
  • Draw a new vector from the origin to the head of the last vector.

• Vectors in Three Dimensions

  • A Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction and can be added to other vectors according to vector algebra.
  • Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on it are all described by vectors.
  • Thus the bound vector represented by $(1,0,0)$ is a vector of unit length pointing from the origin along the positive $x$-axis.
  • The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion.
  • For example, the sum of the vectors $(1,2,3)$ and $(−2,0,4)$ is the vector:

• Tangent Vectors and Normal Vectors

  • A vector is normal to another vector if the intersection of the two form a 90-degree angle at the tangent point.
  • In order for a vector to be normal to an object or vector, it must be perpendicular with the directional vector of the tangent point.
  • When you take the dot product of two vectors, your answer is in the form of a single value, not a vector.
  • Tangent vectors are almost exactly like normal vectors, except they are tangent instead of normal to the other vector or object.
  • These vectors can be found by obtaining the derivative of the reference vector, $\mathbf{r}(t)$:

• Multiplying Vectors by a Scalar

  • Multiplying a vector by a scalar changes the magnitude of the vector but not the direction.
  • A scalar, however, cannot be multiplied by a vector.
  • To multiply a vector by a scalar, simply multiply the similar components, that is, the vector's magnitude by the scalar's magnitude.
  • Most of the units used in vector quantities are intrinsically scalars multiplied by the vector.
  • (i) Multiplying the vector $A$ by the scalar $a=0.5$ yields the vector $B$ which is half as long.

• Unit Vectors and Multiplication by a Scalar

  • In addition to adding vectors, vectors can also be multiplied by constants known as scalars.
  • When multiplying a vector by a scalar, the direction of the vector is unchanged and the magnitude is multiplied by the magnitude of the scalar .
  • Once you have the vector's components, multiply each of the components by the scalar to get the new components and thus the new vector.
  • A useful concept in the study of vectors and geometry is the concept of a unit vector.
  • A unit vector is a vector with a length or magnitude of one.

• Adding and Subtracting Vectors Using Components

  • Another way of adding vectors is to add the components.
  • If we were to add this to another vector of the same magnitude and direction, we would get a vector twice as long at the same angle.
  • This new line is the resultant vector.
  • Vector Addition Lesson 2 of 2: How to Add Vectors by Components
  • This video gets viewers started with vector addition using a mathematical approach and shows vector addition by components.

• The Cross Product

  • The cross product of two vectors is a vector which is perpendicular to both of the original vectors.
  • The result is a vector which is perpendicular to both of the original vectors.
  • Because it is perpendicular to both original vectors, the resulting vector is normal to the plane of the original vectors.
  • The magnitude of vector $c$ is equal to the area of the parallelogram made by the two original vectors.
  • If you use the rules shown in the figure, your thumb will be pointing in the direction of vector $c$, the cross product of the vectors.

• Vectors for Genomic Cloning and Sequencing

  • The four major types of vectors are plasmids, viral vectors, cosmids, and artificial chromosomes.
  • The vector itself is generally a DNA sequence that consists of an insert (transgene) and a larger sequence that serves as the "backbone" of the vector.
  • Simpler vectors called transcription vectors are only capable of being transcribed but not translated: they can be replicated in a target cell but not expressed, unlike expression vectors.
  • Transcription vectors are used to amplify their insert.
  • In the case of plasmids utilized as transcription vectors, incubating bacteria with plasmids generates hundreds or thousands of copies of the vector within the bacteria in hours.

• Calculus of Vector-Valued Functions

  • A vector function is a function that can behave as a group of individual vectors and can perform differential and integral operations.
  • A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors.
  • The input of a vector-valued function could be a scalar or a vector.
  • Vector calculus is a branch of mathematics that covers differentiation and integration of vector fields in any number of dimensions.
  • Because vector functions behave like individual vectors, you can manipulate them the same way you can a vector.