Math Problem Statement

Find a vector equation and parametric equations for the line segment that joins P to Q. P(1, 0, 1), Q(5, 8, 1)

(a) Vector equation. <b>r</b>(t) = langle1+4t, , rangle

(b) Parametric equations. x(t) = 1+4t y(t) =

z(t) =

Solution

To find the vector and parametric equations for the line segment joining points P(1,0,1)P(1, 0, 1) and Q(5,8,1)Q(5, 8, 1), follow these steps:

Step 1: Vector equation

The vector equation of a line is given by:

r(t)=P+t(QP)\mathbf{r}(t) = \mathbf{P} + t(\mathbf{Q} - \mathbf{P})

where:

  • P\mathbf{P} is the position vector of point P(1,0,1)P(1, 0, 1),
  • Q\mathbf{Q} is the position vector of point Q(5,8,1)Q(5, 8, 1),
  • tt is a parameter that varies from 0 to 1 for the line segment.

First, calculate QP\mathbf{Q} - \mathbf{P}: QP=(51,80,11)=(4,8,0)\mathbf{Q} - \mathbf{P} = (5 - 1, 8 - 0, 1 - 1) = (4, 8, 0)

So, the vector equation becomes: r(t)=1,0,1+t4,8,0\mathbf{r}(t) = \langle 1, 0, 1 \rangle + t \langle 4, 8, 0 \rangle

Expanding the vector equation:

r(t)=1+4t,0+8t,1+0t\mathbf{r}(t) = \langle 1 + 4t, 0 + 8t, 1 + 0t \rangle

Thus, the vector equation is:

r(t)=1+4t,8t,1\mathbf{r}(t) = \langle 1 + 4t, 8t, 1 \rangle

Step 2: Parametric equations

From the vector equation r(t)=1+4t,8t,1\mathbf{r}(t) = \langle 1 + 4t, 8t, 1 \rangle, we can write the parametric equations as follows:

x(t)=1+4tx(t) = 1 + 4t y(t)=8ty(t) = 8t z(t)=1z(t) = 1

These parametric equations describe the line segment between points PP and QQ.

Summary:

  • Vector equation:

    r(t)=1+4t,8t,1\mathbf{r}(t) = \langle 1 + 4t, 8t, 1 \rangle

  • Parametric equations:

    x(t)=1+4t,y(t)=8t,z(t)=1x(t) = 1 + 4t, \quad y(t) = 8t, \quad z(t) = 1

Would you like more details on this process, or do you have any questions?

5 Related Questions:

  1. How do you find the direction vector between two points in 3D space?
  2. What is the geometric meaning of parametric equations for a line?
  3. How does the parameter tt control the line segment in parametric equations?
  4. How do vector equations differ when representing lines vs. line segments?
  5. What happens to the parametric equations if points PP and QQ lie on different planes?

Tip:

In parametric equations, t=0t = 0 corresponds to point PP, and t=1t = 1 corresponds to point QQ, which helps limit the line to the segment between these points.

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Math Problem Analysis

Mathematical Concepts

Vector Equations
Parametric Equations
3D Geometry

Formulas

Vector equation of a line: r(t) = P + t(Q - P)
Parametric equations: x(t), y(t), z(t)

Theorems

Parametric equation theorem
Vector subtraction in 3D space

Suitable Grade Level

Grades 10-12