Problem 29 - Advanced Engineering Mathematics

Question

In Problems , determine the points of intersection of the given line and the three coordinate planes. $$ x=4-2 t, y=1+2 t, z=9+3 t $$

Step-by-Step Solution

Answer

The points of intersection are (10, -5, 0), (5, 0, \frac{15}{2}), and (0, 5, 15).

Step 1: Define the Intersection with the XY-plane

For the line to intersect the XY-plane, the z-coordinate must be 0. Use the expression for z: \( z = 9 + 3t = 0 \). Solve this equation to find the value of t that results in this intersection.

Step 2: Solve for t in XY-plane Intersection

Set up the equation: \[ 9 + 3t = 0 \]Solving for t gives:\[ 3t = -9 \]\[ t = -3 \]

Step 3: Calculate Intersection Coordinates in XY-plane

Substitute \( t = -3 \) back into the equations for x and y: x: \( x = 4 - 2(-3) = 4 + 6 = 10 \) y: \( y = 1 + 2(-3) = 1 - 6 = -5 \)Thus, the point of intersection with the XY-plane is \( (10, -5, 0) \).

Step 4: Define the Intersection with the XZ-plane

For the line to intersect the XZ-plane, the y-coordinate must be 0. Use the expression for y: \( y = 1 + 2t = 0 \). Solve this equation to find the value of t.

Step 5: Solve for t in XZ-plane Intersection

Set up the equation: \[ 1 + 2t = 0 \]Solving for t gives:\[ 2t = -1 \]\[ t = -\frac{1}{2} \]

Step 6: Calculate Intersection Coordinates in XZ-plane

Substitute \( t = -\frac{1}{2} \) back into the equations for x and z: x: \( x = 4 - 2\left(-\frac{1}{2}\right) = 4 + 1 = 5 \) z: \( z = 9 + 3\left(-\frac{1}{2}\right) = 9 - \frac{3}{2} = \frac{15}{2} \)Thus, the point of intersection with the XZ-plane is \( (5, 0, \frac{15}{2}) \).

Step 7: Define the Intersection with the YZ-plane

For the line to intersect the YZ-plane, the x-coordinate must be 0. Use the expression for x: \( x = 4 - 2t = 0 \). Solve this equation to find the value of t.

Step 8: Solve for t in YZ-plane Intersection

Set up the equation: \[ 4 - 2t = 0 \]Solving for t gives:\[ 2t = 4 \]\[ t = 2 \]

Step 9: Calculate Intersection Coordinates in YZ-plane

Substitute \( t = 2 \) back into the equations for y and z: y: \( y = 1 + 2(2) = 1 + 4 = 5 \) z: \( z = 9 + 3(2) = 9 + 6 = 15 \)Thus, the point of intersection with the YZ-plane is \( (0, 5, 15) \).

Key Concepts

Points of Intersection

In coordinate geometry, points of intersection are crucial for understanding how objects intersect within the coordinate space. When a line intersects a plane, the intersection is often a single point.
This concept is very important when working on problems involving coordinate planes like XY-plane, XZ-plane, or YZ-plane.
Understanding Intersection on Planes:

• XY-plane: In the XY-plane, the z-coordinate is zero. This means we set the z equation of the line to zero to find the point of intersection.
• Xz-plane: Here, the y-coordinate must be zero, leading to setting the y equation of the line to zero.
• YZ-plane: Similarly, the x-coordinate must be zero, requiring us to use the x equation equivalent to zero.

Grasping these ideas helps in determining where a line crosses different planes in a 3D coordinate system, leading to a better understanding of their spatial relationship.

Parametric Equations

Parametric equations represent lines in a three-dimensional space using parameters, typically denoted as \( t \). These equations express the coordinates \( x \), \( y \), and \( z \) each as functions of \( t \).
For example, the given line is defined by:

• \( x = 4 - 2t \)
• \( y = 1 + 2t \)
• \( z = 9 + 3t \)

This parameter \( t \) allows for a dynamic description of the line, where different values of \( t \) yield different points on the line. Through solving the parametric equations for intersections, you can find the exact point where the line meets the coordinate planes.
It's an efficient way to handle complex spatial relationships in geometry.

Coordinate Planes

Coordinate planes form the foundational elements in three-dimensional geometry. There are three primary coordinate planes:

• XY-plane: It includes all points where the z-coordinate is zero. Think of it as a flat sheet extending indefinitely within the X and Y axis dimensions.
• XZ-plane: This plane includes all points where the y-coordinate is zero, extending across the X and Z axis dimensions.
• YZ-plane: Consists of all points where the x-coordinate is zero, covering the Y and Z axis dimensions.

These planes are crucial for visualizing and solving geometric problems involving spatial relationships.
Knowing how lines intersect these planes will help in mastering three-dimensional geometry concepts.