Problem 1

Question

Give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x=2, y=3$$

Step-by-Step Solution

Verified

Answer

The set of points is a line parallel to the z-axis, where x = 2 and y = 3.

1Step 1: Analyze the equation x = 2

The equation $x = 2$ describes all points in space that have an $x$-coordinate of 2. This represents a plane parallel to the $y\text{-}z$ plane at $x = 2$.

2Step 2: Analyze the equation y = 3

Similarly, the equation $y = 3$ describes all points in space that share the same $y$-coordinate of 3. This plane is parallel to the $x\text{-}z$ plane at $y = 3$.

3Step 3: Intersection of the planes

The intersection of the planes from steps 1 and 2 ($x = 2$ and $y = 3$) is the set of all points with $x = 2$ and $y = 3$. Since they also extend infinitely along the $z$-axis, the intersection is a line parallel to the $z$-axis.

Key Concepts

Coordinate PlanesIntersection of PlanesLines in Space

Coordinate Planes

When we talk about coordinate planes, we refer to two-dimensional surfaces on which points exist in three-dimensional space. A coordinate plane is defined by two of the three axes in a 3D space: the x-axis, y-axis, and z-axis. These planes are:

**xy-plane**: All points that lie on this plane have a z-coordinate of zero. It is formed by the x and y axes.
**xz-plane**: All points on this plane have a y-coordinate of zero. It is formed by the x and z axes.
**yz-plane**: All points on this plane have an x-coordinate of zero. It is formed by the y and z axes.

When a point has a constant value in one of these dimensions, such as in the equation $x = 2$, it represents a plane parallel to one of the primary coordinate planes. In this case, $x = 2$ implies a plane parallel to the yz-plane, shifted to the position where every point has its x-coordinate equal to 2.

Intersection of Planes

Understanding the intersection of planes gives insight into the more complex spatial relationships within 3D geometry. When two planes intersect, they typically form a line. The points of this line are those which satisfy both plane equations concurrently. For example:

In the equations $x = 2$ and $y = 3$, each represents a distinct plane.
The plane $x = 2$ is parallel to the yz-plane, while $y = 3$ is parallel to the xz-plane.

The intersection of these planes will be a line where both the **x** and **y** coordinates are fixed - specifically, $x = 2$ and $y = 3$. The line extends indefinitely along the z-direction, because there are no z-coordinate restrictions.

Lines in Space

Lines in space are one-dimensional and can extend infinitely in both directions in a 3D environment. These lines can be defined in various ways:

As the intersection of two planes, such as with the planes $x = 2$ and $y = 3$.
Through a parametric representation, where a point and a direction vector describe the line.

In the case of intersecting planes $x = 2$ and $y = 3$, the resultant line is fixed along these planes. This line,

is described by parametric equations like $x = 2$, $y = 3$, and $z = t$, where $t$ can be any real number.

This showcases the flexibility and unique descriptive power lines have within the framework of 3D geometry.

Problem 1

Problem 2

Other exercises in this chapter

Problem 1

Find the length and direction (when defined) of $\mathbf{u} \times \mathbf{v}$ and $\mathbf{v} \times \mathbf{u}$. $$\mathbf{u}=2 \mathbf{i}-2 \mathbf{j}-\m

View solution

Problem 1

Let $\mathbf{u}=\langle 3,-2\rangle$ and $\mathbf{v}=\langle-2,5\rangle .$ Find the (a) component form and (b) magnitude (length) of the vector. $$3 \mathbf

View solution

Problem 2

Find parametric equations for the lines. The line through $P(1,2,-1)$ and $Q(-1,0,1)$

View solution

Problem 2

Find a. $\quad \mathbf{v} \cdot \mathbf{u},|\mathbf{v}|,|\mathbf{u}|$ b. the cosine of the angle between $\mathbf{v}$ and $\mathbf{u}$ c. the scalar compo

View solution