Problem 33 - University Calculus: Early Transcendentals

Question

Find the distance from the point (4,3,0) to the a. \(x\) -axis b. \(y\) -axis c. z-axis

Step-by-Step Solution

Answer

Distances are 3, 4, and 5 for x, y, and z axes respectively.

Step 1: Understanding the Problem

We are given a point \(4, 3, 0\) and need to find its distance to the \x\-axis, \y\-axis, and \z\-axis. This involves finding the perpendicular distance from the point to each axis in the 3D coordinate system.

Step 2: Distance to the x-axis

The distance from a point \(x, y, z\) to the \x\-axis is the length of the segment perpendicular to the \x\-axis, which is located on the plane where \x\ equals 4. Therefore, the distance is calculated using the coordinates \y\ and \z\. For the point \(4, 3, 0\), the distance is \sqrt{y^2 + z^2} = \sqrt{3^2 + 0^2} = 3\.

Step 3: Distance to the y-axis

The distance from a point \(x, y, z\) to the \y\-axis is the segment length perpendicular to the \y\-axis, situated on the plane where \y\ equals 3. Use coordinates \x\ and \z\ to calculate this distance. For the given point \(4, 3, 0\), the formula is \sqrt{x^2 + z^2} = \sqrt{4^2 + 0^2} = 4\.

Step 4: Distance to the z-axis

To find the distance from a point \(x, y, z\) to the \z\-axis, take the perpendicular segment from the point to the \z\-axis on the plane where \z\ equals 0. The distance utilizes \x\ and \y\ coordinates. Therefore, for \(4, 3, 0\), it is \sqrt{x^2 + y^2} = \sqrt{4^2 + 3^2} = 5\.

Key Concepts

Perpendicular Distance

In the context of a 3D coordinate system, understanding perpendicular distance is crucial. Imagine a point in space, such as (4, 3, 0), and an axis, like the x-axis. The perpendicular distance from this point to the x-axis is the shortest path between them, forming a right angle with the axis. This is fundamentally different from simply measuring along a straight line.

The concept of perpendicular distance is tied to the Pythagorean theorem. When calculating this distance in a 3D cube, you'll measure along two of the three possible coordinate axes. For instance, the distance from (4, 3, 0) to the x-axis involves the y and z coordinates:

• The x-axis doesn't move vertically or forward/backward, only side-to-side.
• Think of the motion towards the x-axis as going through the yz-plane.

This principle applies similarly to the y and z axes, where the concern is always with the coordinates not belonging to the axis in question. Perpendicular distance provides a realistic way to understand how far a point truly is from an axis, not skewed by angles or other dimensions.

3D Geometry

3D geometry is an exciting step up from the usual 2D plane, adding an extra dimension, which means we think about depth in addition to width and height. In a 3D coordinate system, each point is defined by three coordinates: (x, y, z). This opens up a world of possibilities in spatial reasoning and geometric calculations.

The axes in a 3D system—x, y, and z—are mutually perpendicular. These axes form the foundation of the 3D space, and all points are described in terms of their distance along each of these axes.

• Understanding 3D space is like visualizing a room with length, width, and height.
• The x-axis might stretch from one side of the room to the other, the y-axis from front to back, and the z-axis from floor to ceiling.

This multi-dimensional space requires us to use more complex formulas for calculations. For example, determining the distance between two points or from a point to an axis involves considering how far along each axis the point sits. It's a leap into spatial reasoning that challenges our everyday perception limited to two dimensions.

Coordinate System Axes Distance

To grasp how the coordinate system axes distance works, imagine a typical room with the x, y, and z axes forming the edges. The distance from a point to an axis in a 3D space is effectively the direct perpendicular drop to the axis.

When calculating the distance from a point like (4, 3, 0) to any axis, such as the x-axis:

• The required distance is found by ignoring the coordinate matching the axis (x in this case).
• You then calculate using the remaining coordinates, y and z.

This results in using the Pythagorean Theorem, a fundamental tool in geometry, allowing us to compute distances as:

• Distance to the x-axis: \[\sqrt{y^2 + z^2}\]
• Distance to the y-axis: \[\sqrt{x^2 + z^2}\]
• Distance to the z-axis: \[\sqrt{x^2 + y^2}\]

For our example point (4, 3, 0), this gives precise distances, guiding us to truly understand the proximity of a point to each axis in the three-dimensional plane. Recognizing these distances helps solidify spatial awareness and the mathematical skills available with 3D coordinate geometry.