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 and , follow these steps:
Step 1: Vector equation
The vector equation of a line is given by:
where:
- is the position vector of point ,
- is the position vector of point ,
- is a parameter that varies from 0 to 1 for the line segment.
First, calculate :
So, the vector equation becomes:
Expanding the vector equation:
Thus, the vector equation is:
Step 2: Parametric equations
From the vector equation , we can write the parametric equations as follows:
These parametric equations describe the line segment between points and .
Summary:
-
Vector equation:
-
Parametric equations:
Would you like more details on this process, or do you have any questions?
5 Related Questions:
- How do you find the direction vector between two points in 3D space?
- What is the geometric meaning of parametric equations for a line?
- How does the parameter control the line segment in parametric equations?
- How do vector equations differ when representing lines vs. line segments?
- What happens to the parametric equations if points and lie on different planes?
Tip:
In parametric equations, corresponds to point , and corresponds to point , 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
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