Question

\(P(x, 5, z)\) is on a line through \(Q(2,-4,3)\) that is parallel to one of the coordinate axes. Which axis must it be and what are \(x\) and \(z ?\)

Short answer

The line is parallel to the y-axis, x=2, z=3.

Step-by-step solution

Identifying the Axis

First, determine which axis the line is parallel to. Given that it is parallel to one of the coordinate axes, we should analyze the coordinates of the points Q and P. The line is parallel to an axis if there is no change or only one component changes when moving from Q to P.

Analyzing the y-coordinate

Notice that in point Q(2, -4, 3), the y-coordinate is -4, while in point P(x, 5, z), the y-coordinate is 5. This means there is a change in the y-coordinate. Thus, the line cannot be parallel to the y-axis.

Analyzing the x-coordinate

Since points Q and P have different x and y coordinates, the only possibility is that P and Q differ only by the z-coordinate. Thus, you should check if the x-coordinate has changed. Since x remains undetermined in P(x, 5, z), assume no change.

Determining the Changed Coordinate

To determine which axis the line cannot be parallel to, notice the z in Q is 3, meaning that for P on the axis, the z should remain the same. Given that z remains undetermined in point P, we assume it's unchanged.

Conclusion on Parallel Axis

From the previous analysis, the axis must be the y-axis because both the x and z coordinates would remain the same if unchanged.

Determining x and z Coordinates

Since the line is parallel to the y-axis, the x and z coordinates of P should be the same as for Q, based on the properties of lines parallel to axes. Therefore, set x=2 and z=3 in point P.

Key concepts

Parallel Lines

In coordinate geometry, understanding parallel lines is crucial. Parallel lines are two or more lines that never meet and are equidistant from each other at all points. When a line is parallel to one of the coordinate axes, this means that one of its directional components does not change.

For example, if a line is parallel to the x-axis, its y and z components will remain constant, while the x component could vary. Parallel lines maintain their direction by preserving this consistency in their components, which is particularly relevant in 3D geometry. By identifying that in our exercise, there's a constant shift between Q and P with no other changes, we pinpoint that parallel lines demand a fixed directional component.

Coordinate Axes

The coordinate axes in 3D geometry are the x, y, and z axes. They provide a framework for describing the location of points in space. In our exercise, we are looking at a line that is parallel to one of these axes.

Each axis corresponds to one of the dimensions:

• The x-axis represents movement in the width direction.
• The y-axis represents movement in the height direction.
• The z-axis represents depth.

For a line to be parallel to an axis, the movement along that axis is constant, and changes occur only in the other dimensions. From the exercise, it's clear that the y-coordinate changes, suggesting a parallel alignment with the y-axis, as other coordinates would remain static if unchanged.

3D Geometry

3D geometry extends the concepts of coordinate geometry into three dimensions. It involves the x, y, and z coordinates, which allow us to locate points in space comprehensively.

Understanding 3D geometry is essential for tasks such as identifying whether a line is parallel or if a point lies within a particular plane. Objects in 3D space can move or rotate within these three axes, providing a wide range of possible transformations. From our provided problem, understanding these movements enables deducing the parallel nature of lines in relation to the axes, confirming spatial relations between points like Q and P.

Points in 3D Space

Points in 3D space are characterized by their coordinates \(x, y, z\). Each point is an exact location within the three-dimensional coordinate system. The coordinates determine where the point is situated along the three axes.

In our problem, the point P has undetermined x and z coordinates, which affects how it aligns with Q. To find which axis the line through Q and P is parallel to, observe changes (or lack thereof) in each point's coordinates. Here, the mechanics of points in 3D space are at play in evaluating changes or stasis along each axis, highlighting the significance of each coordinate's role, leading to the discovery that the line is parallel to the y-axis.