Lesson 1: Lines in R2 and R3

• Equation of a line

  The mathematical expression that describes the position of a line in a coordinate plane or space
• Forms of the equation of a line

  • Slope-intercept form (y = mx + b)
  • Standard form (Ax + By + C = 0)
  • Point-slope form (y - y1 = m(x - x1))
  • Vector equation (r = r0 + tm)
  • Parametric equations (x = x0 + tm1, y = y0 + tm2, z = z0 + tm3)
• Determining the equation of a line

  • Requires a point and a direction vector or 2 points
• Finding the vector equation of a line

  1. Let P(x, y) be any point on the line

  2. Write the vector OP as OP0 + t*m, where OP0 is a position vector to a known point P0 and m is the direction vector

  3. The vector equation is r = r0 + tm, where t is a real number parameter
• Finding the parametric equations of a line

  1. Express the components of the vector r as separate equations

  2. x = x0 + tm1, y = y0 + tm2, z = z0 + tm3, where t is a real number parameter