vector area

Examples of vector area in the following topics:

• 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.

• Scalars vs. Vectors

  • Vectors require both a magnitude and a direction.
  • The magnitude of a vector is a number for comparing one vector to another.
  • In the geometric interpretation of a vector the vector is represented by an arrow.
  • Some examples of these are: mass, height, length, volume, and area.
  • An example of a vector.

• 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.
  • A vector field $\mathbf{v}$, whose curl is zero, is called irrotational .
  • Such vortex-free regions are examples of irrotational vector fields.
  • The line integral over a scalar field $f$ can be thought of as the area under the curve $C$ along a surface $z=f(x,y)$, described by the field.

• Physics and Engineering: Fluid Pressure and Force

  • Pressure ($p$) is force per unit area applied in a direction perpendicular to the surface of an object.
  • Mathematically, $p = \frac{F}{A}$, where $p$ is the pressure, $\mathbf{F}$ is the normal force, and $A$ is the area of the surface on contact.
  • It relates the vector surface element (a vector normal to the surface) with the normal force acting on it.
  • The pressure is the scalar proportionality constant that relates the two normal vectors:
  • The subtraction (–) sign comes from the fact that the force is considered towards the surface element while the normal vector points outward.

• Curl and Divergence

  • The curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field.
  • A vector field whose curl is zero is called irrotational.
  • The curl is a form of differentiation for vector fields.
  • If $\hat{\mathbf{n}}$ is any unit vector, the projection of the curl of $\mathbf{F}$ onto $\hat{\mathbf{n}}$ is defined to be the limiting value of a closed line integral in a plane orthogonal to $\hat{\mathbf{n}}$ as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.
  • Divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point in terms of a signed scalar.

• 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, $\mathbf{r}(t)$:

• Matter Exists in Space and Time

  • $(A - B)^2 = A^2 -2AB + B^2$: The areas of plane figures equal the sum of the areas of their parts.
  • Logically a beginning knowledge of vectors, vectors spaces and vector algebra is needed to understand his ideas.
  • Ladder Boom Rescue: Vector analysis is methodological.
  • Every vector has a component and a magnitude-direction form.
  • Newton used vectors and calculus because he needed that mathematics.