E-MAG Lesson 1

• Scalar


  A quantity whose value may be represented by a single (positive or negative) real number
• Vector


  A quantity that has both a magnitude and a direction in space
• Scalar quantities


  • mass, density, pressure (but not force), volume, volume resistivity, and current
• Vector quantities


  • force, velocity, acceleration
• Cartesian Coordinate System


  Also known as the rectangular coordinate system. Axes convention follows the "right-hand rule". A point in space may be described by stating the x, y, and z coordinates.
• Vector Components


  • x-coordinate is equal to the distance of P from the yz plane
  • y-coordinate is equal to the distance of P from the xz plane
  • z-coordinate is equal to the distance of P from the xy plane
• Vector


  Identified by giving its three component vectors each lying along the three coordinate axes, whose vector sum is the given vector
• Vector Addition


  1. Follows the parallelogram law and is commutative and associative

  2. Negating a vector reverses its direction
• Vector Subtraction


  Reversing the direction of a vector and adding it
• Scalar-Vector Multiplication


  1. Obeys associative and distributive laws

  2. A vector reverses its direction when multiplied by a negative scalar

  3. Dividing a vector by a scalar value is just multiplying it with the reciprocal of the scalar
• Vector Addition Identities

  • Commutative
  • Associative
  • Distributive
• Position Vector


  Describes the position of a point in space
• Distance Vector


  Describes the distance between two points in space
• Vector Magnitude (Euclidean Norm)

  • The magnitude of a vector B, written |B| or simply B, is given by the square root of the sum of the squares of its components
  • The unit vector in the direction of B, or B/|B|, is given by dividing the vector by its magnitude
• Dot or Scalar Product

  • The product of the magnitude of A, the magnitude of B, and the cosine of the smaller angle between them
  • Obeys the commutative and distributive laws
  • If the vectors are orthogonal, the dot product is zero
• Cross or Vector Product

  • The magnitude of the cross product is equal to the product of the magnitudes of A and B and the sine of the smaller angle between them
  • The direction of the cross product is perpendicular to the plane containing A and B and is along that one of the two possible perpendiculars which is in the direction of advance of a right-handed screw as A is turned into B
• Cross or Vector Product Properties


  • It is not commutative
  • It is not associative
  • It is distributive
  • Scale property
  • Zero identity
  • Components
• Triple Product


  • Scalar Triple Product: The volume of the rectangular parallelepiped having A, B, and C as the edges
  • Vector Triple Product: The vector obtained by taking the cross product of A with the cross product of B and C
• Scalar Projection


  The component of vector A in the direction of vector B
• Vector Projection


  The vector component of A in the direction of B
• Vector Field


  A vector function of a position vector. A collection of different vectors in magnitude and direction.
• Scalar Field Examples


  • Density at any point P within a volume
  • Elevation of a point (x,y) from sea level
  • Temperature at inside a container
• Vector Field Examples


  • Wind Strength and direction in a region
  • Magnetic Field of the Earth