Problem 42 Find parametric equations of the... [FREE SOLUTION]

Chapter 11: Problem 42

Find parametric equations of the line of intersection of the planes. $$ \begin{aligned} &3 x-5 y+2 z=0 \\ &z=0 \end{aligned} $$

Short Answer

Expert verified

Parametric equations are: $x = t$, $y = \frac{3}{5}t$, $z = 0$.

Step by step solution

Write down equations of the planes

We have two planes given to us by the equations: $3x - 5y + 2z = 0$ and $z = 0$. The first plane is a general plane equation, while the second plane effectively restricts us to the XY-plane.

Substitute the value of z in the first plane

Since $z = 0$, substitute $z = 0$ into the equation of the first plane: $3x - 5y + 2(0) = 0$. This simplifies to $3x - 5y = 0$.

Solve for y in terms of x

From the equation $3x - 5y = 0$, solve for $y$ in terms of $x$: $3x = 5y$, thus $y = \frac{3}{5}x$.

Set up parametric equations for the line

Let $x = t$ be the parameter. Then, based on our relationship $y = \frac{3}{5}x$, we have $y = \frac{3}{5}t$ and since $z = 0$, $z = 0$.

Write the final parametric equations

The parametric equations for the line of intersection are: $x = t$, $y = \frac{3}{5}t$, $z = 0$. These describe a line along the intersection of the two planes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Lines of Intersection

When two planes intersect, they generally do so along a line. To determine this line, we first need to understand how the equations of the two planes interact.

In our exercise, we have two planes: one described by the equation $3x - 5y + 2z = 0$, and the other by $z = 0$.
The second plane set $z$ to zero, restricting the intersection to the XY-plane.
By substituting $z = 0$ into the first plane's equation, we simplify to $3x - 5y = 0$, reducing the dimension and effectively describing the line of intersection.

This intersection forms a line because it is the set of points that satisfy both plane equations simultaneously.

Plane Equations

A plane in three-dimensional space is usually described by a linear equation in terms of $x, y,$ and $z$.

In our case, the first plane is given by $3x - 5y + 2z = 0$.
This is a general form where each coefficient relates to how much each variable contributes to the position in space.
The second plane simplifies matters by defining $z = 0$, meaning it lies completely within the XY-plane. This means any point on this plane must have a $z$ value of 0.

Understanding these equations helps to navigate the relationships between different planes, particularly in spatial geometry.

Parametric Representation

Parametric equations are a powerful way to represent lines or curves. They express the coordinates of the points as functions of a single parameter, often making problems easier to solve.

After reducing our system of equations, we find $y = \frac{3}{5}x$ by substituting $z = 0$ into the plane equation.
We then introduce a parameter $t$ to represent $x$, so $x = t$.
Consequently, $y = \frac{3}{5}t$, and $z = 0$ (from the plane $z = 0$).

Thus, the parametric equations: $x = t$, $y = \frac{3}{5}t$, $z = 0$ succinctly describe the line of intersection, offering a straightforward mathematical representation of the problem's solution.

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91影视

Find parametric equations of the line of intersection of the planes. $$ \begin{aligned} &3 x-5 y+2 z=0 \\ &z=0 \end{aligned} $$

Short Answer

Step by step solution

Write down equations of the planes

Substitute the value of z in the first plane

Solve for y in terms of x

Set up parametric equations for the line

Write the final parametric equations

Key Concepts

Lines of Intersection

Plane Equations

Parametric Representation

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