Problem 3 - Essential Calculus Early Transcendentals

Question

Which of the points \(A(-4,0,-1), B(3,1,-5),\) and \(C(2,4,6)\) is closest to the \(y z\) -plane? Which point lies in the \(x z\) -plane?

Step-by-Step Solution

Answer

Point C(2,4,6) is closest to the yz-plane, and point A(-4,0,-1) lies in the xz-plane.

Step 1: Identifying the y z-plane

To find the point closest to the yz-plane, we need to understand that the yz-plane is defined by all points where the x-coordinate is 0.

Step 2: Calculating Distances to y z-plane

For each point, calculate the absolute value of their x-coordinates, as this represents distance to the yz-plane. For point A(-4,0,-1), it is 4; for point B(3,1,-5), it is 3; and for point C(2,4,6), it is 2.

Step 3: Finding the Closest Point to y z-plane

Compare the distances calculated. The point with the smallest distance to the yz-plane is C(2,4,6), with a distance of 2.

Step 4: Identifying the x z-plane

The xz-plane is defined by all points where the y-coordinate is 0.

Step 5: Checking Points in x z-plane

Identify which point has a y-coordinate of 0. Point A(-4,0,-1) has a y-coordinate of 0, thus it lies in the xz-plane.

Key Concepts

distance to plane

When we talk about the "distance to plane," we refer to how far a particular point is from a designated plane in three-dimensional space. To illustrate, consider the 3D coordinate system that is formed by the x, y, and z axes. Each point has three coordinates, representing its location relative to each of these axes.

To find the distance of a point to a specific plane like the yz-plane, we primarily focus on one coordinate — the x-coordinate, if the yz-plane is our reference. The distance is simply the absolute value of this coordinate:

• For a point \(P(x, y, z)\), its distance to the yz-plane is \(|x|\).
• The closer \(|x|\) is to zero, the nearer the point is to the yz-plane.

Understanding this concept is crucial because it helps in visualizing and calculating spatial relationships, which is a fundamental aspect for students dealing with 3D geometry.

yz-plane

The yz-plane is a fundamental plane in a 3D coordinate system.
It consists of all the points that have an x-coordinate of zero. This means that no matter how far you move along the y or z axes, as long as the x-coordinate stays zero, you're still on the yz-plane.

To find which point is closest to the yz-plane, you need to notice how far points are from having their x-coordinate equal to zero. For example:

• If you have a point \(A(-4, 0, -1)\), the distance to the yz-plane \(\text{is } |x| = 4\).
• Similarly, for a point \(B(3, 1, -5)\), \(|x| = 3\).
• For \(C(2, 4, 6)\), \(|x| = 2\).

Here, the point C is closest to the yz-plane since it has the smallest x-coordinate in absolute value, which is 2.

xz-plane

The xz-plane is the set of all points where the y-coordinate is equal to zero. This plane is useful when you want to visualize movements and positions that ignore changes in the y-direction.

Identifying which points lie in the xz-plane comes down to checking their y-coordinates:

• A point like \(A(-4, 0, -1)\) lies in the xz-plane because its y-coordinate is 0.
• Points with any other y-value lie outside of the xz-plane.

In practical applications, the xz-plane can help simplify problems by reducing the number of dimensions you need to consider. It plays an important role in fields like engineering and physics, where assumptions about planes aid in simplifying complex systems.