How to Convert Vector Form to Parametric Form

Converting the vector form of a line into its parametric form is a straightforward process that involves separating the components of the position and direction vectors. This conversion essentially breaks down a single vector equation into a set of scalar equations, one for each dimension.

Understanding the Forms

Before converting, let's quickly define the two forms for a line in space:

• Vector Form: Represents a line as the sum of a position vector (a point on the line) and a scalar multiple of a direction vector (the direction of the line).
  • General form: r = a + tv
  • Where:
    • r is a generic position vector of any point (x, y, z) on the line.
    • a is the position vector of a known point (x₀, y₀, z₀) on the line.
    • v is the direction vector (v₁, v₂, v₃) of the line.
    • t is a scalar parameter that can take any real value.
• Parametric Form: Represents the coordinates of any point on the line as functions of the scalar parameter t. It's a set of equations, one for each coordinate.
  • General form:
    • x = x₀ + t * v₁
    • y = y₀ + t * v₂
    • z = z₀ + t * v₃

The Conversion Process

The conversion involves equating the corresponding components of the vector form equation. As noted in the reference, you "do that for both the position vector as well as the direction vector. And its parameter." This means you match the x-components, y-components, and z-components on both sides of the vector equation r = a + tv.

Let the position vector a = ⟨x₀, y₀, z₀⟩ and the direction vector v = ⟨v₁, v₂, v₃⟩. The generic point on the line is r = ⟨x, y, z⟩.

The vector equation is:
⟨x, y, z⟩ = ⟨x₀, y₀, z₀⟩ + t⟨v₁, v₂, v₃⟩

First, perform the scalar multiplication (t times v):
⟨x, y, z⟩ = ⟨x₀, y₀, z₀⟩ + ⟨t v₁, t v₂, t * v₃⟩

Next, perform the vector addition (a + tv):
⟨x, y, z⟩ = ⟨x₀ + t v₁, y₀ + t v₂, z₀ + t * v₃⟩

Finally, equate the corresponding components:
x = x₀ + t v₁
y = y₀ + t v₂
z = z₀ + t * v₃

These last three equations are the parametric equations for the line.

Step-by-Step Guide

Here's a simple breakdown of the process:

1. Identify the position vector a = ⟨x₀, y₀, z₀⟩ (a point the line passes through) and the direction vector v = ⟨v₁, v₂, v₃⟩ from the vector equation r = a + tv.
2. Set the x-coordinate of the generic point (x) equal to the x-component of the position vector (x₀) plus the parameter (t) multiplied by the x-component of the direction vector (v₁). This gives you the first parametric equation: x = x₀ + t * v₁.
3. Repeat step 2 for the y-coordinates: y = y₀ + t * v₂.
4. Repeat step 2 for the z-coordinates: z = z₀ + t * v₃.
5. The resulting set of equations is the parametric form of the line.

Example Conversion

Let's convert the following vector equation of a line into parametric form:
r = ⟨1, -2, 4⟩ + t⟨3, 0, -5⟩

1. Identify the vectors:
   • Position vector a = ⟨x₀, y₀, z₀⟩ = ⟨1, -2, 4⟩
   • Direction vector v = ⟨v₁, v₂, v₃⟩ = ⟨3, 0, -5⟩
2. Apply the formula:
   • x = x₀ + t * v₁
   • y = y₀ + t * v₂
   • z = z₀ + t * v₃
3. Substitute the values from the example:
   • x = 1 + t * 3
   • y = -2 + t * 0
   • z = 4 + t * (-5)
4. Simplify the equations:
   • x = 1 + 3t
   • y = -2
   • z = 4 - 5t

The parametric form of the line is:
x = 1 + 3t
y = -2
z = 4 - 5t

This set of equations tells you how to find the coordinates (x, y, z) of any point on the line by choosing a value for the parameter t.

Summary Table

This table summarizes how each component of the vector equation contributes to the corresponding parametric equation.

This method provides a clear and systematic way to translate the compact vector representation of a line into the component-wise equations of the parametric form.