Lecture: Finding Vector, Parametric, and Symmetric Equations of a Line in 3D Space

Key Concepts

• Line Definition: A line is defined by a point on the line and the direction of the line.
• Vector Equation: A vector equation of a line can be formed if either a point and a vector parallel to the line are given, or if two points on the line are given.

Case 1: Given a Point and a Vector Parallel to the Line

Problem Statement

• Given: Point
  • Coordinates: ((x_0, y_0, z_0)) -> (r_0)
• Given: Parallel Vector
  • (\mathbf{v} = (a, b, c))
• Goal: Find the vector equation, parametric equations, and symmetric equations of the line.

Solution Steps

1. Vector Equation

   • Formula: (\mathbf{r} = \mathbf{r_0} + t\mathbf{v})
   • Position Vector: (\mathbf{r_0} = x_0\mathbf{i} + y_0\mathbf{j} + z_0\mathbf{k})
   • Vector: (\mathbf{v} = a\mathbf{i} - b\mathbf{j} + c\mathbf{k})

2. Example

   • Given:
     • Point: ((2, 4, -3))
     • Vector: ((3, -2, 5))
   • Vector Equation: (\mathbf{r} = (2\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}) + t(3\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}))

3. Parametric Equations

   • Formulas:
     • (x = x_0 + at)
     • (y = y_0 - bt)
     • (z = z_0 + ct)
   • Example:
     • (x = 2 + 3t)
     • (y = 4 - 2t)
     • (z = -3 + 5t)

4. Symmetric Equations

   • Formula: (\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c})
   • Example:
     • (\frac{x - 2}{3} = \frac{y - 4}{-2} = \frac{z + 3}{5})

Case 2: Given Two Points on the Line

Problem Statement

• Given: Two points
  • (A(x_1, y_1, z_1))
  • (B(x_2, y_2, z_2))
• Goal: Find the vector equation, parametric equations, and symmetric equations of the line.

Solution Steps

1. Vector Calculation

   • Find vector (\mathbf{AB}) from points A and B
   • (\mathbf{AB} = (x_2 - x_1, y_2 - y_1, z_2 - z_1))

2. Example

   • Given:
     • Points: ((1, 3, -2)) and ((4, 1, 5))
   • Vector (\mathbf{AB}): ((4 - 1, 1 - 3, 5 + 2) = (3, -2, 7))
   • Using Point A for (\mathbf{r_0})

3. Vector Equation

   • (\mathbf{r} = (1\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}) + t(3\mathbf{i} - 2\mathbf{j} + 7\mathbf{k}))

4. Parametric Equations

   • (x = 1 + 3t)
   • (y = 3 - 2t)
   • (z = -2 + 7t)

5. Symmetric Equations

   • (\frac{x - 1}{3} = \frac{y - 3}{-2} = \frac{z + 2}{7})

Summary

• A line in 3D space can be described using vector, parametric, and symmetric equations.
• These equations can be derived either from a point and a parallel vector or from two points on the line.
• The essential formulas used for transforming given data into these equations were discussed and exemplified.