Problem 4 - 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 \(Q\) $$ P(5,0,10), Q(3,-6,7) $$

Step-by-Step Solution

Answer

The distance between points P and Q is 7 units.

Step 1: Plot the Points

To identify the location of points \(P\) and \(Q\), plot them on a 3-dimensional coordinate system. \(P(5,0,10)\) represents a point with \(x=5\), \(y=0\), and \(z=10\). Similarly, \(Q(3,-6,7)\) is plotted with \(x=3\), \(y=-6\), and \(z=7\).

Step 2: Use the Distance Formula

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

Step 3: Substitute the Coordinates

Substitute the coordinates of points \(P(5,0,10)\) and \(Q(3,-6,7)\) into the distance formula: \[ d = \sqrt{(3 - 5)^2 + (-6 - 0)^2 + (7 - 10)^2} \].

Step 4: Calculate the Differences

Calculate the differences for each coordinate: \((x): 3 - 5 = -2\), \((y): -6 - 0 = -6\), \((z): 7 - 10 = -3\).

Step 5: Square Each Difference

Square each of the differences calculated in the previous step: \((-2)^2 = 4\), \((-6)^2 = 36\), \((-3)^2 = 9\).

Step 6: Sum the Squares

Add up these squared values: \(4 + 36 + 9 = 49\).

Step 7: Take the Square Root

Take the square root of the sum: \(\sqrt{49} = 7\).

Step 8: Final Step: Conclusion

The distance between points \(P\) and \(Q\) is 7 units.

Key Concepts

Distance Formula

Understanding the distance formula is crucial when dealing with 3D coordinate geometry. In 3D space, points are represented by three coordinates, \(x, y, z\). To measure the straight line distance between any two points, we use the 3D distance formula. This formula may look complex, but it is an extension of the Pythagorean theorem, only with an extra dimension.

The formula is as follows: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

Let's break it down:

• Subtract the x-coordinates of both points to find the difference in the x-plane, and then square the result.
• Follow the same process for the y-coordinates and z-coordinates. Squaring these differences ensure they are positive.
• Add up all the three squared differences. This sum represents the squared distance in 3D space.
• Finally, take the square root of this sum to get the actual distance between the two points.

With practice, applying this formula can become second nature whenever finding distances in 3D space.

Plotting Points

Plotting points in a 3D coordinate system might seem daunting at first, but it becomes straightforward with a little practice. A 3D coordinate system includes three perpendicular axes: x, y, and z. Points are represented like this: \(x, y, z\).

Here are some steps to help visualize and plot these points:

• Identify the given coordinates. For example, point \({P(5, 0, 10)}\) has x-coordinate 5, y-coordinate 0, and z-coordinate 10.
• Start by plotting each axis on a graph, remembering that the z-axis is usually the vertical line.
• Begin at the origin \(0, 0, 0\)\ and move along the x-axis to 5.
• Next, move parallel to the y-axis until you reach 0, which in this case means staying on the x-plane.
• Finally, move along the z-axis up to 10.

With these steps, you can see where the point lies in space. Repeating this process helps solidify the concept of plotting in 3D, as practiced in the case of points P and Q.

3D Space

3D space adds a whole new dimension, literally, to geometry. As opposed to 2D space, which uses just two axes, 3D includes an additional axis. This makes it possible to describe objects and locations in the real world more precisely.

The three axes commonly used are:

• **X-axis**: Represents width.
• **Y-axis**: Represents depth.
• **Z-axis**: Represents height or elevation.

These axes meet at the origin \(0, 0, 0\)\ and are usually depicted in diagrams as perpendicular lines forming a "corner" in space. Understanding and visualizing this setup is essential for solving problems in 3D geometry. It provides insights into how distances are calculated, points are plotted, and shapes are understood. Each coordinate tells you how far along each axis a point is from the origin, helping you "see" the point in 3D.

Whether for mathematical computations or real-world applications, mastering 3D space forms a foundational skill in various fields such as physics, engineering, and computer graphics.