Problem 3 - Algebra and Trigonometry

Question

\(3-6\) Two points \(P\) and \(Q\) are given. (a) Plot \(P\) and \(Q .(b)\) Find the distance between \(P\) and \(Q\) $$ P(3,1,0), Q(-1,2,-5) $$

Step-by-Step Solution

Answer

The distance between points P and Q is approximately 6.48.

Step 1: Understand the Coordinates

Identify and write down the coordinates for points \( P \) and \( Q \): \( P(3, 1, 0) \) and \( Q(-1, 2, -5) \). These coordinates are in three-dimensional space, meaning each point is defined by an \( x \), \( y \), and \( z \) component.

Step 2: Plot Points on a 3D Graph

Using graph plotting software or graph paper designated for 3D plotting, mark the point \( P \) at (3,1,0) and point \( Q \) at (-1,2,-5). In 3D space, the x-axis is represented horizontally, the y-axis is vertically, and the z-axis is perpendicular to both. Ensure each point is correctly aligned according to its coordinates.

Step 3: Use the Distance Formula

The distance \( d \) between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in 3D space is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] For points \( P(3,1,0) \) and \( Q(-1,2,-5) \), substitute the values into the formula.

Step 4: Substitute the Coordinates

Substituting the coordinates of \( P \) and \( Q \) into the distance formula: \[ d = \sqrt{((-1) - 3)^2 + (2 - 1)^2 + ((-5) - 0)^2} \] Simplifying inside the square roots gives: \[ d = \sqrt{(-4)^2 + 1^2 + (-5)^2} \]

Step 5: Calculate the Squares

Calculate the squares: \[ (-4)^2 = 16, \quad 1^2 = 1, \quad (-5)^2 = 25 \] Thus, substituting these into the formula gives: \[ d = \sqrt{16 + 1 + 25} \]

Step 6: Find the Total Sum under the Square Root

Summing these values under the square root gives: \[ d = \sqrt{42} \]

Step 7: Calculate the Square Root

Finally, calculate \( \sqrt{42} \), which is approximately 6.48. Thus, the distance between points \( P \) and \( Q \) is \( 6.48 \).

Key Concepts

3D Coordinate System

In the 3D coordinate system, each point is represented by three values: x, y, and z. These values are frequently called coordinates and describe the position of a point in space. The 3D coordinate system is an extension of the 2D system most are familiar with, adding a third dimension to illustrate depth.

• The x-axis runs horizontally, just like on a 2D graph. It's often thought of as the east-west axis.
• The y-axis runs vertically, similar to the 2D case, representing the north-south direction.
• The z-axis is perpendicular to the x and y axes. Usually, this axis represents the up-down direction, moving out of the page or screen in physical models.

By using these three axes, any point in space can be uniquely defined with its coordinates. For example, the point \( P(3, 1, 0) \) indicates a location 3 units along the x-axis, 1 unit along the y-axis, and exactly on the origin plane of the z-axis (because the z-coordinate is 0). Understanding these axes forms the groundwork of navigating and plotting in 3D space.

Plotting Points

To plot the points in a 3D coordinate system, one must understand how each of the x, y, and z coordinates affect a point's position in space. Unlike 2D plotting, which uses just a flat surface, 3D plotting grapples with a third dimension.
When plotting a point such as \( P(3, 1, 0) \), start first with the x-coordinate, moving 3 units along the x-axis. Next, adjust vertically by moving 1 unit up along the y-axis. Lastly, ensure the point stays at the z coordinate, in this case, it stays on the x-y plane as the z value is 0. Repeat a similar process for other points.
Visualizing or sketching with graph software helps make this concept more tangible:

• Use proper scaling for each axis to ensure accuracy.
• Check that the orientation of your axes matches conventional positioning: x is horizontal, y is vertical, and z pops out towards you.
• Imagine or employ grid lines to see how each coordinate fits into the 3D space.

Learning to plot in 3D is essential for solving spatial problems and is useful in various fields requiring spatial analysis.

Square Root Calculation

Calculating the square root is often a key step in various mathematical operations, such as finding the distance between two points. The square root of a number is a value that, when multiplied by itself, gives the original number. In 3D space, the distance between two points A \((x_1, y_1, z_1)\) and B \((x_2, y_2, z_2)\) involves computing the square root as follows: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] This formula derives from extending the Pythagorean theorem into three dimensions. Each squared term accounts for the distance along each respective axis, collectively giving the Euclidean distance between the points.
The steps to calculate are straightforward:

• First, subtract coordinates to find the differences \((x_2 - x_1), (y_2 - y_1), (z_2 - z_1)\).
• Next, square each of these differences.
• Sum the squared values.
• Finally, find the square root of this sum to get the distance.

For example, the distance between points \( P(3,1,0) \) and \( Q(-1,2,-5) \) is calculated as \( \sqrt{42} \), which approximates to 6.48.