Question

Find the distance between the two points. $$ (4,1,5),(8,2,6) $$

Short answer

The distance between the points (4,1,5) and (8,2,6) is approximately 4.24 units.

Step-by-step solution

Calculate differences between coordinates

Calculate the differences between the x, y, and z coordinates of the two points. This will give us: \(x_2 - x_1 = 8 - 4 = 4\), \(y_2 - y_1 = 2 - 1 = 1\) and \(z_2 - z_1 = 6 - 5 = 1\).

Square and sum the differences

Now we square these differences and add them together: \((8 - 4)^2 + (2 - 1)^2 + (6 - 5)^2 = 16 + 1 + 1 = 18 \).

Square root of the sum

Compute the square root of the sum obtained in the previous step: \(\sqrt{18} \approx 4.24\) (rounded to two decimal places).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves studying geometric shapes and figures using a coordinate system. This system helps in defining the positions of points in a space, which can be two-dimensional (2D) or three-dimensional (3D).

In the context of finding the distance between two points in three-dimensional space, coordinate geometry simplifies the problem by allowing us to consider the dimensions independently. We start by identifying the coordinates of the two points, which in our exercise are represented as \( (4,1,5) \) and \( (8,2,6) \). Each number in these ordered triples represents a point's position along the x, y, and z axes, respectively.

Understanding Coordinates

Each point in 3D space is defined by three coordinates. For instance, the point \( (4,1,5) \) indicates that the point is 4 units along the x-axis, 1 unit up the y-axis, and 5 units out along the z-axis. Similarly, the point \( (8,2,6) \) is 8 units along the x-axis, 2 units up the y-axis, and 6 units out along the z-axis.

Using these coordinates, we can deduce the relative positions of points to one another and perform operations such as finding distances or angles between them, making coordinate geometry an essential tool in various applications.

Three-Dimensional Space

Three-dimensional space is an extension of the two-dimensional plane we are familiar with from algebra and geometry. It incorporates depth in addition to height and width, allowing for a more comprehensive representation of the physical world.

When we transition from 2D to 3D, we add an additional coordinate axis, typically known as the z-axis, which is perpendicular to both the x and y axes. You can envision these axes as intersecting lines at a point called the origin, where each axis is labeled with positive and negative directions.

Navigating Through 3D Space

To find the distance between two points in three-dimensional space, as we do with our exercise, we extend the concept of the straight-line distance from two dimensions. We consider the movement along each axis separately, and then we combine these movements to find the straight-line ('as-the-crow-flies') distance.

The process involves figuring out how 'far' we have to travel from one point to the other along each axis. This is done by subtracting the coordinates of the first point from the coordinates of the second point, yielding the steps taken along the x, y, and z axes to get from one point to the other.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, traditionally associated with two-dimensional space. It states that for a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is commonly written as \( a^2 + b^2 = c^2 \).

In three-dimensional space, we can apply it by considering the diagonal between two points as the hypotenuse of a right-angled triangle, whose sides are parallel to the axes of the coordinate plane.

Applying Pythagoras in 3D

Our exercise requires us to use this theorem in a three-dimensional context. In this case, we form right-angled triangles along the three dimensions. We first calculate the 'legs' of these triangles by finding the differences between the corresponding coordinates, then squaring these differences, and finally, we sum them up, as shown in the step-by-step solution. The result represents the square of the diagonal's length. Taking the square root of this sum gives us the distance between the two points, which in essence, is the length of the hypotenuse in 3D space.