Problem 10 - Calculus an Applied Approach

Question

Find the coordinates of the point. $$ \begin{array}{l}{\text { The point is located in the } y z \text { -plane, three units to the right }} \\ {\text { of the } x z \text { -plane, and two units above the } x y \text { -plane. }}\end{array} $$

Step-by-Step Solution

Answer

The coordinates of the point are (0, 3, 2).

Step 1: Identify the Plane

According to the exercise, the point is located in the yz-plane. By definition, any point that lies in the yz-plane has an x-coordinate of 0.

Step 2: Find the Y Coordinate

The exercise states that the point is 'three units to the right of the xz-plane'. In a standard 3D coordinate system, moving 'to the right' of the xz-plane means moving along the positive y-axis. Thus, the y-coordinate of the point is 3.

Step 3: Find the Z Coordinate

The exercise also states that the point is 'two units above the xy-plane'. In a standard 3D coordinate system, moving 'above' the xy-plane means moving along the positive z-axis. Therefore, the z-coordinate of the point is 2.

Key Concepts

YZ-Plane

In a 3D coordinate system, the yz-plane is crucial for locating points in space. Imagine a flat, vertical surface extending infinitely along the y-axis and z-axis, while penetrating through the origin. This plane is where the x coordinate is always zero.
This is because the x-axis is perpendicular to the yz-plane, making the yz-plane the space where the x-component doesn't affect the position of any point.

• If a point is in the yz-plane, its x-coordinate will definitely be zero.
• Such a point relies solely on its y and z coordinates to define its location.

Understanding this helps us solve equations or identify locations by dismissing the x variable when focusing exclusively on the yz-plane.

XY-Plane

The xy-plane is another fundamental component of the 3D coordinate system. It's like a flat tabletop that extends infinitely along the x-axis and y-axis, but lies flat with the z-axis cut straight through it.

In this plane, any point has a z-coordinate of zero, indicating that the point lies completely flat on the xy surface without any elevation.

• When a point is above or below this plane, it has a positive or negative z value, respectively.
• For points lying exactly on the plane, the z-component is zero, providing a two-dimensional view along the x and y axes.

Recognizing when a point is "above" or "below" the xy-plane can help in understanding its spatial positioning in relation to the other planes.

XZ-Plane

The xz-plane serves as a critical plane to visualize in the 3D coordinate system's arena. This plane is like a flat wall, extending along the x and z axes, meeting at the y-axis's zero point. Any point on this plane will have a y-coordinate of zero.

Picture the xz-plane as a wall standing tall on the x and z grid, with unrestricted space to the left and right.

• Points in this plane are influenced only by their x and z positions.
• Knowing that the y value is zero simplifies computations when dealing with this plane.

This understanding can aid students in determining directions and distances since any movement away from this plane will affect the y-coordinate, increasing or decreasing its value depending on direction.