MODULE 1

• Scalar


  Quantities with magnitude only
• Vector


  Quantities with magnitude and direction
• Tensor


  Considered as a multidimensional array of numbers, which are known as its scalar components or simply components
• Complex scalar


  Also referred to as phasor
• Field


  A function of a vector which connect an arbitrary origin to any point in space
• Collinear Vectors


  Vectors that lie on the same line or parallel lines
• Coplanar Vectors


  Vectors lying on same plane
• Notation used in vector analysis


  • Ԧ𝐴, 𝐵, Ԧ𝐶, etc. - vectors (or scalars)
  • 𝑎𝐴, 𝑎𝐵, 𝑎𝐶 - unit vectors
  • 𝑎𝑥, 𝑎𝑦, 𝑎𝑧 - unit vectors in the direction of x, y, and z axes
  • 𝑅𝐴𝐵, 𝑅𝐵𝐶, 𝑅𝑆𝑃 - vector from one point to another
  • 𝑟𝐴, 𝑟𝐵, 𝑟𝐶 - vector from the origin to a point

  
• Vector in space


  Using the Cartesian coordinates, a vector in space is drawn as an ARROW, where its LENGTH is the magnitude and ARROWHEAD is the direction

  
• When a vector is drawn as an arrow of finite length, its location is defined to be at the tail end of the arrow
• A coordinate system specifies a point uniquely in a plane by a pair (triple) of numerical coordinates, which are the signed distances from fixed perpendicular lines

  
• Plotting in 2D and 3D Cartesian Coordinate System

  • 2D Cartesian Coordinate System
  • 3D Cartesian Coordinate System

  
• Distance formula in three dimensions

  The distance |𝑃1𝑃2| between the points 𝑃1 𝑥1, 𝑦1, 𝑧1 and P2 𝑥2, 𝑦2, 𝑧2 is 𝑷𝟏𝑷𝟐 = 𝒙𝟐 − 𝒙𝟏 𝟐 + 𝒚𝟐 − 𝒚𝟏 𝟐 + 𝒛𝟐 − 𝒛𝟏 𝟐
• Equation of a sphere


  An equation of a sphere with center C(h, k, l) and radius r is: 𝒙 − 𝒉 𝟐 + 𝒚 − 𝒌 𝟐 + 𝒛 − 𝒍 𝟐 = 𝒓𝟐
• Problem solving: Example 1

  Find an equation of the sphere that passes through the point (4 3, -1) and has a center (3, 8, 1)
• Problem solving: Example 2


  Show that the equation 𝑥2 + 𝑦2 + 𝑧2 + 4𝑥 − 6𝑦 + 2𝑧 + 6 = 0 represents a sphere, and find its center and radius

  
• Vector in space


  The vector is described as: 𝑨 = 𝑨𝒙𝒂𝒙 + 𝑨𝒚𝒂𝒚 + 𝑨𝒛𝒂𝒛 where: 𝐴𝑥𝑎𝑥, 𝐴𝑦𝑎𝑦, 𝐴𝑧𝑎𝑧 - vector components of Ԧ𝐴, 𝐴𝑥, 𝐴𝑦, 𝐴𝑧 - scalar components in the direction of x-, y-, and z-axis, respectively, 𝑎𝑥, 𝑎𝑦, 𝑎𝑧- unit vectors in the direction of x-, y-, and z-axis, respectively
• Magnitude and direction of vector


  The magnitude of the vector is denoted as 𝐴 and mathematically expressed as: 𝑨 = 𝑨𝒙𝟐 + 𝑨𝒚𝟐 + 𝑨𝒛𝟐
• Unit vectors


  Unit vector is the direction of a vector and denoted as 𝑎𝐴, 𝑎𝐵, 𝑎𝐶 and mathematically expressed as: 𝒂𝑨 = 𝑨 𝑨 = 𝑨𝒙𝒂𝒙 + ��𝒚𝒂𝒚 + 𝑨𝒛𝒂𝒛 𝑨𝒙𝟐 + 𝑨𝒚𝟐 + 𝑨𝒛𝟐
• Direction angles


  The direction angles of a nonzero vector are the three angles that have the smallest nonnegative radian measures 𝛼, 𝛽, 𝛾 measured from the positive x, y, and z axes, respectively, to the position representation of the vector. It is mathematically expressed as: 𝒄𝒐𝒔𝜶 = 𝑨𝒙 𝑨 ; 𝒄𝒐𝒔𝜷 = 𝑨𝒚 𝑨 ; 𝒄𝒐𝒔𝜸 = 𝑨𝒛 𝑨

  
• Types of vectors


  • Positional Vector - a vector from origin to a point
  • Displacement Vector - a vector from one point to another

  
• Problem solving: Example 3

  Find the magnitude and direction cosines of the vector A = <3, 2, -6>

  
• Positional Vector


  A vector from origin to a point

  
• Displacement Vector


  A vector from one point to another
• Finding the magnitude and direction cosines of a vector


  1. Magnitude

  2. Direction cosines

  
• Finding the distance vector and its magnitude from one point to another


  1. Distance vector

  2. Magnitude
• Vector addition


  • Follows the triangle or parallelogram law
  • Can be done by drawing both vectors from the origin and completing the parallelogram
  • Or by beginning the second vector from the head of the first vector and completing the triangle
• Adding three or more vectors graphically


  • Join vectors one at a time
• Vector subtraction


  Reverse the sign of the second vector then add the two vectors
• Vector multiplication by scalars


  • Magnitude of the vector changes while the direction does not change if the scalar is positive
  • Direction changes if the scalar is negative
• Laws on vector algebra

  • Commutative law for addition and multiplication
  • Associative law for addition and multiplication
  • Distributive property
• Finding the magnitude of a vector


  |a| = √(𝑎𝑥^2 + 𝑎𝑦^2 + 𝑎𝑧^2)
• Vector notation


  <x, y> denotes a vector

  (x, y) denotes a point

  <x, y> denotes a vector from origin to (x, y)

  |R| is the magnitude of vector R

  θ is the direction angle measured from the x-axis

  
• Resolving a vector into its components

  𝐴𝑥 = 𝐴 cos 𝜃

  𝐴𝑦 = 𝐴 sin 𝜃
• Resultant


  A vector that has the same effect on a body, in both translation and rotation, as all the original vectors combined
• Coplanar concurrent force system


  • Forces meet at a common point and are on the same plane
• Finding the x and y components of a displacement vector


  𝑅𝑥 = 𝑅 cos 𝜃

  𝑅𝑦 = 𝑅 sin 𝜃

  
• Concurrent


  Forces meet a common point
• Displacement vector R


  Magnitude of R = 175 m, points at angle of 50° relative to the x-axis
• Displacement vector A


  145 m in a direction 20° east of north