he distance between 2 points (x1,Y1) and (x2,y2) is given by d = square root (x1-x2)^2 + (y1-y2)^2

a - pick 2 arbitrary points in 3 dimensions, (x1, y1, and Z1) and (x2,y2,z2) and plot these points. Not that there are 90 degrees between the xy, xz, and yz axes.

b- using your sketch from part a, find the distance between these 2 points, generalize the above equation to 3 dimension

a) To plot two arbitrary points in three dimensions and illustrate the 90 degrees between the XY, XZ, and YZ axes, follow these steps:

1. Assign values to the coordinates (x1, y1, z1) and (x2, y2, z2) of the two arbitrary points. For example, let's say (x1, y1, z1) = (1, 2, 3) and (x2, y2, z2) = (4, 5, 6).

2. Draw a three-dimensional coordinate system with X, Y, and Z axes intersecting at point (0, 0, 0). Make sure the axes are perpendicular to each other, forming a 90-degree angle between them.

3. Locate and label the point (1, 2, 3) on the graph. This point lies in the positive direction of the x-axis by 1 unit, positive direction of the y-axis by 2 units, and positive direction of the z-axis by 3 units.

4. Similarly, locate and label the point (4, 5, 6) on the graph. This point lies in the positive direction of the x-axis by 4 units, positive direction of the y-axis by 5 units, and positive direction of the z-axis by 6 units.

5. Connect the two points with a straight line segment, representing the distance between them.

b) To find the distance between the two points in three dimensions and generalize the distance formula:

1. Use the same points as mentioned in part a: (x1, y1, z1) = (1, 2, 3) and (x2, y2, z2) = (4, 5, 6).

2. The distance between these two points is given by the formula:
d = √((x1 - x2)^2 + (y1 - y2)^2 + (z1 - z2)^2)

3. In this case, the formula becomes:
d = √((1 - 4)^2 + (2 - 5)^2 + (3 - 6)^2)
= √((-3)^2 + (-3)^2 + (-3)^2)
= √(9 + 9 + 9)
= √27
= 3√3

So, the distance between the two points (1, 2, 3) and (4, 5, 6) is 3√3 units.