Understanding Locus in 3D Space

The locus of a point is the set of all points that satisfy a given condition or set of conditions. In a three-dimensional coordinate system, a point P is represented by its coordinates (x, y, z). These coordinates specify the point's position relative to the x, y, and z axes.

The question asks for the locus of a point P(x, y, z) that moves such that its z-coordinate is always equal to 7. The condition is given by the equation \(z = 7\).

Analyzing the Equation \(z = 7\)

The equation \(z = 7\) places a restriction only on the z-coordinate of any point P(x, y, z) that is part of the locus. It states that no matter where the point is located, its distance from the xy-plane (which is measured along the z-axis) must always be 7 units. The values of x and y for this point are not restricted by this equation.

• The x-coordinate can take any real value.
• The y-coordinate can take any real value.
• The z-coordinate must always be exactly 7.

So, we are looking for all points (x, y, 7) where x and y can be any real numbers.

Geometric Interpretation of \(z = \text{constant}\)

Let's consider what this means geometrically in three dimensions.

• The xy-plane is defined by the equation \(z = 0\).
• The xz-plane is defined by the equation \(y = 0\).
• The yz-plane is defined by the equation \(x = 0\).

An equation involving only one variable in 3D space typically represents a plane parallel to the plane formed by the other two axes.

• \(x = \text{constant}\) represents a plane parallel to the yz-plane.
• \(y = \text{constant}\) represents a plane parallel to the xz-plane.
• \(z = \text{constant}\) represents a plane parallel to the xy-plane.

Why \(z = 7\) Represents a Plane Parallel to the xy-plane

The equation \(z = 7\) specifies that the z-coordinate of every point on the locus is fixed at 7. Since x and y can be any real numbers, the points satisfying the condition are of the form \((x, y, 7)\). This set of points forms a flat surface extending infinitely in the x and y directions, located at a constant height of 7 units above the xy-plane (if we assume the positive direction of the z-axis is upwards). This surface is a plane.

Since all points on this surface have the same z-coordinate, the surface is parallel to the xy-plane, which is the plane where all points have a z-coordinate of 0.

Comparing with Other Loci

Let's consider what the other options would represent:

• Line parallel to x-axis: For a line parallel to the x-axis, both the y and z coordinates must be constant. For example, the locus of points satisfying \(y = c_1\) and \(z = c_2\) (where \(c_1\) and \(c_2\) are constants) is a line parallel to the x-axis.

• Line parallel to y-axis: For a line parallel to the y-axis, both the x and z coordinates must be constant. For example, the locus of points satisfying \(x = c_1\) and \(z = c_2\) is a line parallel to the y-axis.

• Line parallel to z-axis: For a line parallel to the z-axis, both the x and y coordinates must be constant. For example, the locus of points satisfying \(x = c_1\) and \(y = c_2\) is a line parallel to the z-axis.

In our problem, only the z-coordinate is fixed (\(z = 7\)). The x and y coordinates are free to vary. This corresponds to a plane, not a line.

Conclusion

The locus of a point P(x, y, z) such that \(z = 7\) is the set of all points \((x, y, 7)\) where x and y can be any real numbers. This set of points forms a plane that is parallel to the xy-plane and is located at a distance of 7 units from it along the z-axis.

Revision Table: Loci in 3D Space

Understanding basic loci helps identify shapes in 3D geometry.

Additional Information: Coordinate Planes and Axes

In the Cartesian coordinate system for 3D space, we have three mutually perpendicular axes: the x-axis, the y-axis, and the z-axis. These axes intersect at the origin (0, 0, 0).

• The xy-plane is the plane containing the x and y axes. Every point on the xy-plane has a z-coordinate of 0. Its equation is \(z = 0\).
• The xz-plane is the plane containing the x and z axes. Every point on the xz-plane has a y-coordinate of 0. Its equation is \(y = 0\).
• The yz-plane is the plane containing the y and z axes. Every point on the yz-plane has an x-coordinate of 0. Its equation is \(x = 0\).

A plane parallel to one of these coordinate planes will have an equation where the corresponding coordinate is fixed to a constant value, not necessarily zero.