In this video, we are going to talk about how to find the distance between two points and understand the formula. So let’s begin.

If we want to find the distance between the points (1, 2, -5) and (4, 6, 7), what formula do we use? The distance formula used is as follows:

The distance is equal to the square root of the difference between the x-values squared, the difference between the y-values squared, and the difference between the z-values squared.

For the first point, (1, 2, -5), we assign x1, y1, and z1 as 1, 2, and -5, respectively. And for the second point, (4, 6, 7), we assign x2, y2, and z2 as 4, 6, and 7.

Now, let’s calculate the distance.

For x, we have 4 – 1 = 3.

For y, we have 6 – 2 = 4.

For z, we have 7 – (-5) = 7 + 5 = 12.

Calculating the squares: 3^2 = 9, 4^2 = 16, and 12^2 = 144.

Adding the values: 144 + 16 + 9 = 169.

Taking the square root of 169 gives us the distance, which is 13.

So, the distance between the two points (1, 2, -5) and (4, 6, 7) is 13 in three dimensions (x, y, z).

Now, let’s plot the two points to understand visually how to find the distance between them.

We have the x-axis, y-axis, and z-axis.

For the first point (1, 2, -5), we go below in the negative z-direction. We travel one unit along the x-axis and draw a parallel line along the y-axis starting from (0, 2, 0). We then go down approximately 5 units along the z-direction. Let’s call this point P1.

For the second point (4, 6, 7), we move along the x-axis to 4 and then draw a parallel line from there to the y-axis. We draw a line parallel to the x-axis from this point and highlight the point of intersection. From this point, we draw a line parallel to the z-axis and go up 7 units. Let’s call this point P2.

To find the distance between these two points visually, we observe the changes along each axis.

On the x-axis, we travel from 1 to 4, a distance of 3.

On the y-axis, we travel from 2 to 6, a distance of 4.

On the z-axis, we travel from -5 to 7, a distance of 12.

These distances form a right triangle. We draw lines parallel to the y-axis and x-axis, forming a parallelogram. Since it’s a right triangle, using the Pythagorean theorem, we can determine that the hypotenuse is 13.

Now, let’s explain this using variables. Let’s say the first point is the origin, and we travel x units along the x-axis, y units along the y-axis, and z units parallel to the z-axis. We can draw a right triangle, calling the hypotenuse L. So x^2 + y^2 = L^2.

We can also draw a triangle to find the distance between P1 and P2, which we call d. This forms another right triangle, with L, z, and d as sides. So L^2 + z^2 = d^2.

By substituting L^2 with x^2 + y^2, we get x^2 + y^2 + z^2 = d^2. Taking the square root of both sides gives us the distance formula.

If P1 was not the origin, x and y would be the differences in the x and y values of P1 and P2. So the formula becomes the square root of (x2 – x1)^2 + (y2 – y1)^2 + (z2 – z1)^2.

This is how the distance formula can be derived using geometry and right triangles.