Problem 56 - Thomas Calculus

Question

In Exercises \(53-56,\) find the point in which the line meets the plane. $$ x=-1+3 t, \quad y=-2, \quad z=5 t ; \quad 2 x-3 z=7 $$

Step-by-Step Solution

Answer

The intersection point is (-4, -2, -5).

Step 1: Understand the Problem

We need to find the point of intersection between a line and a plane. The line is given by the parametric equations \( x = -1 + 3t \), \( y = -2 \), and \( z = 5t \). The plane has the equation \( 2x - 3z = 7 \).

Step 2: Substitute Line Equations into the Plane Equation

Substitute the expressions for \( x \) and \( z \) from the line into the plane's equation. This gives:\[2(-1 + 3t) - 3(5t) = 7.\]

Step 3: Simplify the Equation

Simplify the equation from Step 2:\[-2 + 6t - 15t = 7.\]Combine like terms to get:\[-9t - 2 = 7.\]

Step 4: Solve for the Parameter \( t \)

Solve the equation for \( t \):\[-9t = 7 + 2 \-9t = 9 \t = -1.\]

Step 5: Find the Intersection Point Coordinates

Now that we have \( t = -1 \), substitute this value back into the parametric equations of the line to find the coordinates of the intersection point:\(x = -1 + 3(-1) = -4, \y = -2, \z = 5(-1) = -5.\) Hence, the intersection point is \((-4, -2, -5)\).

Step 6: Verify the Solution

Substitute the coordinates \((-4, -2, -5)\) into the plane's equation to verify:\[2(-4) - 3(-5) = -8 + 15 = 7.\] The plane's equation is satisfied, confirming that the intersection point is correct.

Key Concepts

Parametric Equations

Parametric equations are a fundamental way to represent a line in three-dimensional space. They define the coordinates of any point on the line in terms of a single variable, often called the parameter, denoted as \(t\). This method of line representation is particularly useful when dealing with problems involving intersections between lines and surfaces like planes. In the given exercise, the line is represented in parametric form as follows:

• \(x = -1 + 3t\)
• \(y = -2\)
• \(z = 5t\)

This means that for every specific value of \(t\), there corresponds a point \((x, y, z)\) on the line. Each component equation gives the coordinate of the point in terms of \(t\), allowing us to substitute and solve for intersection with the plane.The power of parametric equations lies in their ability to easily describe complex paths and geometries, making them essential in both mathematics and applied sciences.

Intersection Point

The intersection point is where the line meets the plane. To find this point, we need to solve for the parameter \(t\) that satisfies both the line's parametric equations and the plane equation. Once \(t\) is determined, it can be plugged back into the line's equations to find the specific coordinates of the intersection.To find \(t\), we substitute the expressions for \(x\) and \(z\) from the parametric equations into the plane's equation:\[ 2(-1 + 3t) - 3(5t) = 7 \]After simplifying and solving, we find \(t = -1\). With \(t\) known, we substitute back into:

• \(x = -1 + 3(-1) = -4\)
• \(y = -2\)
• \(z = 5(-1) = -5\)

Thus, the intersection point is \((-4, -2, -5)\). Finding the intersection point involves careful substitution and solving, which is crucial in many real-world applications like computer graphics and collision detection in physics simulations.

Plane Equation

A plane in three-dimensional space can be described by an equation of the form \(Ax + By + Cz = D\), where \(A\), \(B\), and \(C\) are constants that determine the plane's orientation, and \(D\) is another constant. The given exercise uses the plane equation:\[ 2x - 3z = 7 \]This equation states that there is a continuous set of points \((x, y, z)\) satisfying this condition, forming a flat surface in 3D space. For solving the intersection problem, we focus on the substitutes for coordinates from the line into this equation.The absence of \(y\) in this plane equation indicates that the plane is parallel to the \(y\)-axis in the \(x-z\) plane. As a result, the value of \(y\) from the parametric equation directly carries over to the intersection point.Understanding the orientation and characteristics of the plane helps in visualizing and solving geometry-based problems. Plane equations are pivotal in fields such as computer-aided design (CAD) and geographic information systems (GIS), where accurate spatial representations are needed.