Problem 35 - Precalculus Mathematics for Calculus

Question

A line has parametric equations $$x=2+t, \quad y=3 t, \quad z=5-t$$ and a plane has equation \(5 x-2 y-2 z=1\) (a) For what value of \(t\) does the corresponding point on the line intersect the plane? (b) At what point do the line and the plane intersect?

Step-by-Step Solution

Answer

(a) \(t = 1\); (b) Point: \((3, 3, 4)\).

Step 1: Understand Parametric Equations of the Line

The parametric equations given are \(x=2+t\), \(y=3t\), and \(z=5-t\). These describe a line in 3D space where \(t\) is the parameter.

Step 2: Plane Equation

The plane is given by the equation \(5x - 2y - 2z = 1\). This is a standard linear equation that represents a 2D plane in 3D space.

Step 3: Substitute Parametric Equations into Plane Equation

Substitute the parametric expressions for \(x\), \(y\), and \(z\) into the plane equation. Start with: \[5(2+t) - 2(3t) - 2(5-t) = 1\].

Step 4: Simplify the Expression

Expand and simplify the equation from Step 3:\[10 + 5t - 6t - 10 + 2t = 1\]Combine like terms:\[t = 1\].

Step 5: Solve for t

From step 4, the equation \(t = 1\) indicates that the line intersects the plane when \(t\) equals 1.

Step 6: Find the Intersection Point

Substitute \(t = 1\) back into the parametric equations of the line to find the point of intersection:\(x = 2 + 1 = 3\), \(y = 3(1) = 3\), \(z = 5 - 1 = 4\).The intersection point is \((3, 3, 4)\).

Key Concepts

Parametric Equations

Parametric equations are a powerful tool in geometry, especially when dealing with lines. They allow us to express the coordinates of points on a line using a parameter, often denoted as "\(t\)". Each variable, such as \(x\), \(y\), and \(z\), is given as a function of this parameter.

For instance, the equations \(x = 2 + t\), \(y = 3t\), and \(z = 5 - t\) describe a line in three-dimensional space. Here, \(t\) represents a set of values that control the position along the line:

• \(x = 2 + t\): Moving the line parallel to the \(x\)-axis through shifts governed by \(t\).
• \(y = 3t\): This means that the \(y\) value changes directly with \(t\), indicating a proportional change.
• \(z = 5 - t\): Here, the \(z\) value decreases as \(t\) increases, showing an opposite direction relationship.

By varying \(t\), we get different coordinates on the line, each forming a unique point along this path.

Understanding parametric equations of lines is essential as they provide a straightforward way to handle line equations in space, crucial for solving intersection problems like lines intersecting planes or other geometric figures.

3D Geometry

Three-dimensional geometry extends the concepts of 2D geometry into a space where we can navigate using three axes: \(x\), \(y\), and \(z\). When dealing with 3D spaces, it is crucial to visualize the relationships and positions of figures such as lines and planes.

In 3D geometry, a line is typically represented using parametric equations like those mentioned earlier, while a plane is defined using a linear equation of three variables, such as \(5x - 2y - 2z = 1\) in this case. This type of equation characterizes a flat surface extending infinitely in the 3D space.

Visualizing the intersection of a line and a plane involves imagining a scenario where the path of the line pierces the flat surface of the plane. When they intersect, they share a common point that satisfies both the parametric line equations and the plane equation. Finding this point involves substituting the parametric expressions of the line into the plane's equation and solving for \(t\), as demonstrated in the solution steps.

By mastering these concepts, students can better understand how geometric figures relate within a three-dimensional space, an essential skill in fields like physics, engineering, and computer graphics.

Linear Equations in Three Variables

In the context of 3D geometry, linear equations involving three variables (often \(x\), \(y\), and \(z\)) are used to describe planes. The standard form of a plane equation appears like this: \(Ax + By + Cz = D\), where \(A\), \(B\), \(C\), and \(D\) are constants.

In our problem, the equation \(5x - 2y - 2z = 1\) is such a linear equation. Here's what each part means:

• \(5x\): Represents the effect of the \(x\)-coordinate; this coefficient scales the \(x\)-axis contributions to the plane.
• \(-2y\): This adjusts the contribution from the \(y\)-coordinate, pointing in the opposite direction due to the negative sign.
• \(-2z\): Similarly scales and negates the contribution of the \(z\)-coordinate.
• The \(= 1\): Defines where the plane sits in relation to the origin. It specifies a particular "height" or displacement in the 3D space.

To determine the intersection point, we substitute the parametric expressions of the line into the plane's equation to find the precise \(t\) at which the line meets the plane.

This process of solving can be visualized as adjusting \(t\) until the line "touches" the plane, revealed by a valid solution to the equation. These principles extend to solving more complex systems, crucial for multi-variable calculus and linear algebra.