Distance Between Points P(10, 25, -50) And Q(33, 9, 17)

by Alex Johnson 56 views

Have you ever wondered how to calculate the distance between two points in a three-dimensional space? It's a common problem in various fields, from computer graphics to physics. In this article, we'll break down the steps to find the distance between two points, P(10, 25, -50) and Q(33, 9, 17). So, let's dive in and make this concept crystal clear!

Understanding the Distance Formula

When you want to find the distance between two points in a three-dimensional space, you'll need to use the distance formula. This formula is essentially an extension of the Pythagorean theorem, which you might remember from geometry class. The distance formula in 3D is given by:

√((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

Where:

  • (x₁, y₁, z₁) are the coordinates of the first point (P).
  • (x₂, y₂, z₂) are the coordinates of the second point (Q).

This formula might look a bit intimidating at first, but don't worry! We'll break it down step by step. Think of it as calculating the length of the straight line that connects the two points in space. The formula calculates the difference in each dimension (x, y, and z), squares them, adds them up, and then takes the square root. This gives us the straight-line distance, which is exactly what we're looking for.

Now, let’s talk a bit more about why this formula works. Imagine each dimension (x, y, z) as an axis in a 3D coordinate system. The difference in the x-coordinates (x₂ - x₁) represents how far apart the points are along the x-axis. Similarly, the differences in the y and z coordinates represent the distances along the y and z axes, respectively. By squaring these differences, we ensure that we're dealing with positive values, regardless of the order in which we subtract the coordinates. This is crucial because distance is always a non-negative quantity. Adding the squared differences gives us the square of the distance, and taking the square root brings us back to the actual distance.

This formula is not just a mathematical curiosity; it has practical applications in numerous fields. For example, in computer graphics, it's used to calculate distances between objects in a virtual scene, which is essential for rendering and collision detection. In physics, it's used to calculate the distance between particles or objects, which is crucial for understanding their interactions. In navigation, it's used to calculate the distance between two locations, which is fundamental for mapping and route planning. So, understanding this formula is a valuable skill that can be applied in many different contexts.

Applying the Formula to Points P and Q

Now, let's apply the distance formula to our specific points, P(10, 25, -50) and Q(33, 9, 17). This is where the fun begins! We'll take the coordinates of these points and plug them into the formula we discussed earlier. This process involves a bit of arithmetic, but if we take it one step at a time, it's quite manageable.

First, we identify the coordinates:

  • For point P, we have x₁ = 10, y₁ = 25, and z₁ = -50.
  • For point Q, we have x₂ = 33, y₂ = 9, and z₂ = 17.

Now, we substitute these values into the distance formula:

√((33 - 10)² + (9 - 25)² + (17 - (-50))²)

See? It’s just a matter of plugging in the numbers. Next, we'll simplify the expression inside the square root. This involves performing the subtractions within the parentheses, squaring the results, and then adding them all together. It's like following a recipe – each step builds upon the previous one.

Let's start with the first set of parentheses: (33 - 10) = 23. We then square this result: 23² = 529. This represents the squared difference in the x-coordinates.

Next, we move on to the second set of parentheses: (9 - 25) = -16. Squaring this gives us (-16)² = 256. This is the squared difference in the y-coordinates. Remember that squaring a negative number always results in a positive number, which is important for our distance calculation.

Finally, we tackle the last set of parentheses: (17 - (-50)) = 17 + 50 = 67. Squaring this gives us 67² = 4489. This represents the squared difference in the z-coordinates.

Now that we have the squared differences in each dimension, we add them together: 529 + 256 + 4489 = 5274. This is the sum of the squared differences, which is a crucial step in calculating the distance.

So, our formula now looks like this:

√(5274)

We're almost there! The final step is to take the square root of this sum to find the distance between the two points. This will give us the final numerical answer.

Calculating the Distance

After applying the formula, we've simplified the problem to finding the square root of 5274. This is the final step in determining the distance between points P and Q. You might reach for a calculator at this point, and that's perfectly fine! Calculating square roots of large numbers by hand can be time-consuming and prone to errors. Using a calculator ensures accuracy and efficiency.

So, let's calculate the square root of 5274. When you do this, you'll find that:

√5274 ≈ 72.62

Therefore, the distance between points P(10, 25, -50) and Q(33, 9, 17) is approximately 72.62 units. It's important to remember that this is an approximate value, as the square root of 5274 is an irrational number with a non-repeating, non-terminating decimal expansion. We've rounded the result to two decimal places for practical purposes.

This result gives us a concrete measure of how far apart these two points are in three-dimensional space. Imagine a straight line connecting point P and point Q; the length of that line is approximately 72.62 units. This understanding of distance is crucial in various applications, from mapping and navigation to computer graphics and physics.

Now, let's think about the units for a moment. We haven't specified any particular units for the coordinates (e.g., meters, feet, inches). Therefore, the distance we've calculated is in the same units as the coordinates. If the coordinates were given in meters, then the distance would be 72.62 meters. If they were given in feet, the distance would be 72.62 feet, and so on.

Understanding the units is essential for interpreting the result correctly. A distance of 72.62 meters has a different meaning than a distance of 72.62 inches, so it's crucial to be mindful of the context in which the distance is being used.

Conclusion

In this article, we've walked through the process of finding the distance between two points in three-dimensional space. We started with the distance formula, which is a powerful tool derived from the Pythagorean theorem. We then applied this formula to the specific points P(10, 25, -50) and Q(33, 9, 17). By carefully substituting the coordinates into the formula and performing the necessary calculations, we found that the distance between these points is approximately 72.62 units.

We also discussed the importance of understanding the units and the practical applications of this calculation in various fields. Whether you're working with computer graphics, physics, or navigation, the ability to calculate distances between points is a fundamental skill.

So, the next time you encounter a problem involving distances in 3D space, remember the distance formula and the steps we've outlined in this article. With a little practice, you'll become a pro at calculating distances in no time!

For further reading on the distance formula and its applications, you might find it helpful to explore resources on Khan Academy's Geometry Section, which offers comprehensive explanations and practice exercises.