Problem 55 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The point of intersection is (1, 1, 0).

Step 1: Represent the Line Parametrically

The line is given parametrically as:\[x = 1 + 2t, \quad y = 1 + 5t, \quad z = 3t\]These equations represent the coordinates \((x, y, z)\) of any point on the line as a function of the parameter \(t\).

Step 2: Write the Equation of the Plane

The equation of the plane is given by:\[x + y + z = 2\]Any point \((x, y, z)\) that satisfies this equation lies on the plane.

Step 3: Substitute Parametric Equations into Plane Equation

Substitute \(x = 1 + 2t\), \(y = 1 + 5t\), and \(z = 3t\) into the plane equation:\[(1 + 2t) + (1 + 5t) + 3t = 2\]Simplify this equation to solve for \(t\).

Step 4: Simplify and Solve for t

Simplify the equation:\[1 + 2t + 1 + 5t + 3t = 2 \2 + 10t = 2\]Subtract 2 from both sides:\[10t = 0\]Solve for \(t\):\[t = 0\]

Step 5: Find the Intersection Point Using t

Substitute \(t = 0\) back into the parametric equations:\[x = 1 + 2(0) = 1\y = 1 + 5(0) = 1\z = 3(0) = 0\]The point of intersection is \((1, 1, 0)\).

Key Concepts

Parametric Equations

Parametric equations are powerful tools used to represent lines in space. They express each coordinate of a point on a line as a function of a parameter, typically represented by \( t \). In this exercise, the parametric equations given are:

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

These equations describe how the coordinates \( (x, y, z) \) change as the parameter \( t \) varies. Each value of \( t \) corresponds to a different point on the line. Thus, by adjusting \( t \), we can trace the entire line in three-dimensional space.
This makes parametric equations very versatile, as they can easily describe both straight lines and curves, depending on how the equations are formulated. In problems involving intersections, substituting these parametric expressions into another equation, such as a plane's equation, helps us find specific points of interest, like where the line meets the plane.

Equation of a Plane

The equation of a plane in three-dimensional space can be imagined as a flat surface extending infinitely in two directions. Given as \( x + y + z = 2 \) in this exercise, this equation defines all points \( (x, y, z) \) that lie on the plane.
The general form of a plane's equation is \( ax + by + cz = d \), where \( a, b, \) and \( c \) are coefficients that determine the plane's orientation, while \( d \) factors in its position relative to the origin. The coefficients \( a, b, \) and \( c \) represent the plane's normal vector, a perpendicular direction to the plane’s surface.
Understanding this equation is vital when determining intersections between a line and a plane. If a point satisfies the plane's equation, it lies on the plane. Thus, by substituting parametric equations of a line into the plane equation, as shown in the original solution, we effectively check which points on the line are also on the plane.

Solving Linear Equations

Solving linear equations involves finding values for variables that satisfy the equation. Here, when the parametric line equations are plugged into the plane's equation, we arrive at a simplified linear equation in terms of \( t \):
\( 2 + 10t = 2 \)
To solve for \( t \), first isolate the variable by subtracting 2 from both sides, resulting in an equation:
\( 10t = 0 \)
Dividing both sides by 10, we find \( t = 0 \). This value of \( t \) tells us the specific point along the line that intersects with the plane.

• The substitution step converts a spatial problem into a simpler algebraic equation.
• The solution \( t = 0 \) is crucial for finding the line-plane intersection point.
• By inserting \( t = 0 \) back into the parametric equations, the precise coordinates \((1, 1, 0)\) of the intersection are obtained.

This step-by-step approach makes it easier to systematically find solutions in problems involving geometry and algebra.