Question

Find the distance from \((6,3,-6)\) to \((10,0,6)\).

Short answer

The distance between the points (6,3,-6) and (10,0,6) is 13 units.

Step-by-step solution

Identify the coordinates of the two points

The problem provides us with the coordinates of the two points in 3-dimensional space. The first point has coordinates (6,3,-6) and the second point has coordinates (10,0,6).

Plug the coordinates into the distance formula

Using the distance formula \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\), plug in the coordinates of both points: \(Distance = \sqrt{(10-6)^2 + (0-3)^2 + (6-(-6))^2}\)

Calculate the difference for each coordinate

Simplify the formula by calculating the difference for each coordinate: \(Distance = \sqrt{(4)^2 + (-3)^2 + (12)^2}\)

Square the differences for each coordinate

Now square the differences for each coordinate: \(Distance = \sqrt{16 + 9 + 144}\)

Add the squared differences

Add the squared differences to get the final value inside the square root: \(Distance = \sqrt{169}\)

Calculate the square root

The final step is to compute the square root of the sum: \(Distance = 13\) The distance between the points (6,3,-6) and (10,0,6) is 13 units.

Key concepts

3-dimensional space

3-dimensional space is where we live and observe the physical world; it incorporates length, width, and height. Think of it as the world we interact with daily, where objects have depth and can exist anywhere in this space. It's often represented with three axes labeled as the x-axis, y-axis, and z-axis. These axes allow us to describe any point in space using three numbers, known as coordinates. For example, an object at

• (x, y, z) = (6, 3, -6)

means it is located 6 units along the x-axis, 3 units along the y-axis, and -6 units along the z-axis.
Visualizing 3-dimensional space goes beyond flat surfaces. Imagine flying above a city. You see the expanse of the city (width), the distance from one end to the other (length), and the height of buildings.
This concept is crucial in fields like physics, engineering, and computer graphics where positions and distances in space are essential.

coordinate geometry

Coordinate geometry, also known as analytic geometry, spatially describes geometric figures using numbers and algebra. This approach allows us to use coordinates to define points, lines, and shapes. By employing algebraic equations, we can precisely calculate distances, angles, and areas.
Let's take a look at the coordinates (6,3,-6) and (10,0,6). These numbers tell us the precise location of each point in 3D space. Using the distance formula, we can calculate how far apart these two points are. This formula is an extension of geometry into algebra, making it possible to solve complex problems with clarity.
With coordinate geometry, we can

• transform shapes
• find midpoints
• determine intersections of lines and planes

Thus, it's a bridge between pure geometry and its application in numerous scientific disciplines.

Pythagorean theorem

The Pythagorean theorem is a basic principle in geometry. It relates the lengths of the sides of a right-angled triangle. According to the theorem, in a right triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides. Expressed as:\[c^2 = a^2 + b^2\]In 3-dimensional space, we apply this theorem to calculate the distance between two points. We extend the formula to include a third coordinate z, modifying the theorem slightly:\[Distance = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]This extended version maps perfectly onto the problem of finding the distance from point (6,3,-6) to (10,0,6). Here, instead of a flat surface, we consider a 3D space.
By breaking down the differences along each axis (x, y, z) and squaring each, the theorem provides us a robust method to find distances in a three-dimensional setup, making it invaluable for spatial calculations.