Problem 30 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

Intersection points: XY-plane (-3, -8, 0), XZ-plane (7/3, 0, 14/3), YZ-plane (0, -7/2, 3).

Step 1: Analyze the Given Equation

The equation given is \[ \frac{x-1}{2} = \frac{y+2}{3} = \frac{z-4}{2} = k \] for some parameter \(k\). This indicates the parametric equations of a line.

Step 2: Find Intersection with the XY-Plane

In the XY-plane, \(z = 0\). Setting \(z = 0\) in the parametric equation, we have:\[ k = \frac{z-4}{2} \Rightarrow k = \frac{0-4}{2} = -2 \]Plug \(k = -2\) into the other equations:\[ x = 1 + 2(-2) = -3 \]\[ y = -2 + 3(-2) = -8 \]The intersection point with the XY-plane is \((-3, -8, 0)\).

Step 3: Find Intersection with the XZ-Plane

In the XZ-plane, \(y = 0\). Setting \(y = 0\) in the parametric equation, we get:\[ k = \frac{y+2}{3} \Rightarrow k = \frac{0+2}{3} = \frac{2}{3} \]Plug \(k = \frac{2}{3}\) into the other equations:\[ x = 1 + 2\left(\frac{2}{3}\right) = \frac{7}{3} \]\[ z = 4 + 2\left(\frac{2}{3}\right) = \frac{14}{3} \]The intersection point with the XZ-plane is \(\left(\frac{7}{3}, 0, \frac{14}{3}\right)\).

Step 4: Find Intersection with the YZ-Plane

In the YZ-plane, \(x = 0\). Setting \(x = 0\) in the parametric equation, we have:\[ k = \frac{x-1}{2} \Rightarrow k = \frac{0-1}{2} = -\frac{1}{2} \]Plug \(k = -\frac{1}{2}\) into the other equations:\[ y = -2 + 3\left(-\frac{1}{2}\right) = -\frac{7}{2} \]\[ z = 4 + 2\left(-\frac{1}{2}\right) = 3 \]The intersection point with the YZ-plane is \(\left(0, -\frac{7}{2}, 3\right)\).

Key Concepts

Parametric Equations

Parametric equations are a powerful way to describe geometric objects, such as curves and surfaces, that vary over a parameter. In this case, the parameter is represented by \(k\), which defines the position on the line in a three-dimensional space.
Each component of the line's position—\(x\), \(y\), and \(z\)—is expressed in terms of this parameter:

• For \(x\), we have \(x = 2k + 1\)
• For \(y\), it is \(y = 3k - 2\)
• And for \(z\), we have \(z = 2k + 4\)

By plugging different values of \(k\) into these equations, one can obtain different points on the line. Parametric equations offer flexibility because they reveal both the line's direction and any specific points on it.
This structure is handy for calculating the intersections of lines with coordinate planes.

Coordinate Planes

Coordinate planes are parts of the three-dimensional Cartesian coordinate system, which consists of three planes:

• The XY-plane is defined by setting \(z = 0\)
• The XZ-plane is where \(y = 0\)
• Lastly, the YZ-plane is given when \(x = 0\)

Each plane serves as a flat surface that divides the 3D space into different sections. To find where a line intersects one of these planes, you need to use the parametric equations of the line and substitute the required plane's condition (like \(z = 0\) for the XY-plane).
By doing this, you isolate the parameter \(k\), enabling you to find the point of intersection by substituting \(k\)'s value back into the equations for \(x\), \(y\), and \(z\). Understanding these planes is key for a range of applications, from computer graphics to physics simulations.

Line Intersections

Line intersections occur where a line crosses another line or a plane. To determine where a line intersects the coordinate planes, like the XY, XZ, and YZ planes, apply the condition that defines each plane:

• To find the intersection with the XY-plane, use \(z = 0\)
• For the XZ-plane, apply \(y = 0\)
• For the YZ-plane, use \(x = 0\)

Once these conditions are applied within the parametric equations, solve for \(k\) and use it to find the corresponding values of \(x\), \(y\), and \(z\).
This will yield the precise intersection points, providing valuable insights into the spatial configuration of the line with respect to the planes. Recognizing how these intersections occur is crucial for real-world navigation and spatial analysis.