vector

Examples of vector in the following topics:

• 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, $\mathbf{r}(t)$:

• 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.

• 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.

• Vectors in the Plane

  • Vectors are needed in order to describe a plane and can give the direction of all dimensions in one vector equation.
  • The plane determined by this point and vector consists of those points $P$ , with position vector $\mathbf{r}$, such that the vector drawn from $P_0$ to $P$ is perpendicular to $\mathbf{n}$.
  • Recall that two vectors are perpendicular if and only if their dot product is zero.
  • The vectors $\mathbf{V}$ and $\mathbf{W}$ can be visualized as vectors starting at $\mathbf{R_0}$ and pointing in different directions along the plane.
  • Calculate the directions of the normal vector and the directional vector of a reference point

• Vector-Valued Functions

  • A vector function covers a set of multidimensional vectors at the intersection of the domains of $f$, $g$, and $h$.
  • Also called vector functions, vector valued functions allow you to express the position of a point in multiple dimensions within a single function.
  • The input into a vector valued function can be a vector or a scalar.
  • In Cartesian form with standard unit vectors (i,j,k), a vector valued function can be represented in either of the following ways:
  • This is a three dimensional vector valued function.

• Vector Fields

  • A vector field is an assignment of a vector to each point in a subset of Euclidean space.
  • In vector calculus, a vector field is an assignment of a vector to each point in a subset of Euclidean space.
  • The elements of differential and integral calculus extend to vector fields in a natural way.
  • A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point.
  • A gravitational field generated by any massive object is a vector field.

• Conservative Vector Fields

  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • A conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential.
  • Therefore, the curl of a conservative vector field $\mathbf{v}$ is always $0$.
  • A vector field $\mathbf{v}$, whose curl is zero, is called irrotational .
  • Such vortex-free regions are examples of irrotational vector fields.

• Surface Integrals of Vector Fields

  • Consider a vector field $\mathbf{v}$ on $S$; that is, for each $\mathbf{x}$ in $S$, $\mathbf{v}(\mathbf{x})$ is a vector.
  • The surface integral of vector fields can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.
  • Alternatively, if we integrate the normal component of the vector field, the result is a scalar.
  • where $r$ is the position vector and $\hat{r}$ is a unit vector in radial direction.
  • Explain relationship between surface integral of vector fields and surface integral of a scalar field

• Equations of Lines and Planes

  • A line is a vector which connects two points on a plane and the direction and magnitude of a line determine the plane on which it lies.
  • The position vector of point $P_0$ is called $\mathbf{r}_0$ and the position vector of point $P$ is called $\mathbf{r}$.
  • The vector from $P$ to $P_0$ is called vector $\mathbf{a}$.
  • Vectors $\mathbf{a}$ and $\mathbf{v}$ are parallel to each other.
  • where $t$ represents the location of vector $\mathbf{r}$ on plane $M$.