Problem 8 - Calculus an Applied Approach

Question

Find the coordinates of the point. $$ \begin{array}{l}{\text { The point is located seven units in front of the } y z \text { -plane, two }} \\ {\text { units to the left of the } x z \text { -plane, and one unit below the }} \\ {x y \text { -plane. }}\end{array} $$

Step-by-Step Solution

Answer

The coordinates of the point are (7, -2, -1).

Step 1: Identify the x-coordinate

The point is the unit distance 'seven units in front of the yz plane'. This distance along the positive x-axis is from the yz-plane, meaning the x-coordinate is at \(x=7\).

Step 2: Identify the y-coordinate

The point is 'two units to the left of xz plane'. This distance along the negative y-axis is from the xz-plane, meaning the y-coordinate is at \(y=-2\).

Step 3: Identify the z-coordinate

The point is 'one unit below the xy plane'. This distance along the negative z-axis is from the xy-plane, meaning the z-coordinate is at \(z=-1\).

Key Concepts

yz-plane

The yz-plane is an essential concept in 3D coordinate geometry. It serves as a reference plane in the three-dimensional space, defined by two axes: the y-axis and the z-axis. In this coordinate system, the x-axis is perpendicular to the yz-plane.

In simpler terms, any point located on the yz-plane has an x-coordinate of zero. This is because the yz-plane consists of all points where the x-value is zero. For a point described as 'seven units in front of the yz-plane', it indicates a position along the x-axis. Specifically, if a point is seven units away from the yz-plane, then its x-coordinate is 7.

• The yz-plane is like a giant wall right at x = 0.
• To find out how far a point is from this plane, you just need to check its x-coordinate.
• For example, the x-coordinate tells us how far left or right of the yz-plane a point is.

Keep in mind, the further the x-value from zero, the further away from the yz-plane the point is positioned. This approach can help visualize and understand the arrangement of points in 3D space.

xz-plane

When discussing the xz-plane, imagine it as a flat surface lying on the horizontal plane at y = 0. The xz-plane is defined by the x-axis and the z-axis, and it's perpendicular to the y-axis. Any point on the xz-plane will have a y-coordinate of zero.

When a point is described as being 'two units to the left of the xz-plane', this refers to its y-coordinate. A leftward position relative to the xz-plane implies a negative value on the y-axis. Thus, a coordinate two units to the left of the xz-plane will have a y-value of -2.

• The xz-plane functions like a floor extending infinitely at y = 0.
• To determine how far a point is from this plane, examine its y-coordinate.
• This means the y-coordinate can tell us how high up or down from the xz-plane a point is located.

Understanding the xz-plane our simplification of 3D visual space, aiding in plotting points and understanding spatial relationships.

xy-plane

The xy-plane is like a vast horizontal plane that extends infinitely at z = 0. Governed by the x-axis and y-axis, this plane serves as a foundation in 3D geometry, where the z-axis stands vertical to it. Points lying on this plane have a z-coordinate of zero.

A position described as 'one unit below the xy-plane' tells us about the placement of a point along the z-axis. Here, the z-coordinate becomes crucial. If a point is one unit below the xy-plane, its z-coordinate is -1 since it's located below or downwards from this plane.

• The xy-plane acts as a vast sheet spread across z = 0.
• To figure out how far a point is from this plane, observe its z-coordinate.
• The z-coordinate reveals how high above or below the xy-plane a point resides.

By understanding these spatial dynamics, one can better grasp how points and planes interact within the realm of 3D space.