Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the distance from \((2,3,-1)\) to (a) the \(x y\)-plane, (b) the \(y\)-axis, and (c) the origin.

Step-by-Step Solution

Answer

(a) 1, (b) √5, (c) √14

Step 1: Understanding the Point and the Coordinate Planes

We need to find distances related to the point \((2,3,-1)\). The coordinates are labeled as \(x = 2, y = 3, z = -1\). The \(xy\) plane is defined where \(z = 0\). Thus, the distance to the plane depends only on the \(z\)-coordinate.

Step 2: Calculating Distance to the xy-plane

The distance from a point to the \(xy\)-plane is the absolute value of the \(z\)-coordinate, since the \(xy\)-plane is \(z = 0\). For the point \((2,3,-1)\), the distance is \(|-1| = 1\).

Step 3: Understanding the y-axis

The \(y\)-axis is defined where both \(x = 0\) and \(z = 0\). To find the distance from a point to the \(y\)-axis, compute the length of the vector formed by the \(x\) and \(z\) coordinates.

Step 4: Calculating Distance to the y-axis

The formula for the distance to the \(y\)-axis is \( ext{Distance} = \sqrt{x^2 + z^2}\). For \((2, 3, -1)\), the distance is \(\sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}\).

Step 5: Understanding the origin and Calculating Distance

To find the distance from a point to the origin \( (0,0,0) \), use the distance formula: \( \sqrt{x^2 + y^2 + z^2} \). For the point \((2,3,-1)\), the distance is \(\sqrt{2^2 + 3^2 + (-1)^2} = \sqrt{4 + 9 + 1} = \sqrt{14}\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that allows us to represent geometric figures using coordinates. This field combines geometry and algebra to study the position of points and lines in space.
In 3D coordinate geometry, each point is represented by three numbers:

• the x-coordinate, which shows how far the point is along the x-axis,
• the y-coordinate, which represents the point's position along the y-axis,
• the z-coordinate, indicating the point's position along the z-axis.

Understanding these coordinates helps us explore distances and relationships between points in space.

Distance Formula

The distance formula is a mathematical equation used to determine the distance between two points in space. This formula originates from the Pythagorean theorem, which can be applied to the 3D coordinate system.
To find the distance between two points \(x_1, y_1, z_1\) and \(x_2, y_2, z_2\), we use the formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
This equation helps us calculate various distances, such as the distance to the \(xy\) plane or to an axis, by setting the corresponding coordinates to zero where necessary.
For example, to find the distance from a point to the origin \( (0,0,0) \), you use: \[ \sqrt{x^2 + y^2 + z^2} \]
This method ensures accurate measurements in a three-dimensional space.

Vectors

Vectors are mathematical objects that have both a magnitude and a direction. In 3D space, vectors are used to represent points or displacements from one point to another.
A vector is often depicted as \( \mathbf{v} = \langle x, y, z \rangle \), which describes movement from the origin to the point \( (x, y, z) \).

• Magnitude refers to the length of the vector, calculated using the distance formula.
• Direction is the way in which the vector points in space.

Vectors are essential when calculating distances to lines or planes. For example, determining the distance from a point to the \(y\)-axis involves finding the vector's magnitude formed by the \(x\) and \(z\) components.

3D Coordinates

3D coordinates extend the concept of coordinates to include a third dimension, providing depth to our understanding of space. This enhances our ability to describe points in a more comprehensive manner.
Unlike 2D coordinates, which only deal with an x and y-axis, 3D coordinates introduce a z-axis. Adding this third dimension allows us to model and visualize spaces and shapes that are not confined to flat surfaces.
The point \( (2, 3, -1) \) serves as an example, where each number corresponds to its position relative to the x, y, and z axes.

• The x-coordinate is 2 units along the x-axis,
• the y-coordinate is 3 units along the y-axis,
• and the z-coordinate is -1 unit along the z-axis.

Understanding 3D coordinates is crucial when solving problems such as finding distances to the \(xy\)-plane, axes, or the origin.