Derive the distance formula (d) shown below for points A = (x1, y1, z1) and

D = (x2, y2, z2).
2
2 1
2
2 1
2
d = (x2 − x1) + (y − y ) + (z

To derive the distance formula for points A = (x1, y1, z1) and D = (x2, y2, z2), we will use the Pythagorean theorem in three dimensions.

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). In our case, we will consider each coordinate (x, y, and z) as a side of the triangle.

Let's begin by drawing a right-angled triangle with the points A and D as two vertices. We can label the third vertex as B.

We can then calculate the lengths of the sides of the triangle.

The length of side AB is the difference between the x-coordinates:
AB = x2 - x1

The length of side BC is the difference between the y-coordinates:
BC = y2 - y1

The length of side AC is the difference between the z-coordinates:
AC = z2 - z1

According to the Pythagorean theorem, the length of the hypotenuse AD is given by:
AD = sqrt(AB^2 + BC^2 + AC^2)

Simplifying the equation by squaring the terms:
AD^2 = (x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2

Taking the square root of both sides, we get the distance formula:
AD = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

Therefore, the distance formula (d) for points A = (x1, y1, z1) and D = (x2, y2, z2) is:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

Remember to substitute the actual coordinates of A and D to calculate the distance between them.