The distance between 2 points (z1,Y1) and (x2,y2) is given by d = square root (x1-x2)+ (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 0- 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 dimensions.
first, the axes are perpendicular, the there are 90°, not 0° between axes.
Second, can't do plots here.
Third, once you have picked your points, just plug them into the 3D version of the formula:
d^2 = (x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2
To see that this is true, consider just the distances in the plane z=z1.
w^2 = (x2-x1)^2+(y2-y1)^2
That's just a diagonal line of length w. To get from (x1,y1,z1) to (x2,y2,z2), consider the right triangle from (x1,y1,z1) to (x2,y2,z1) with length w, and go from there to (x2,y2,z2) of length (z2-z1).
If that's confusing, just look online for references to 3D distances.
http://tonto.stanford.edu/~brian/euclidean_distance_in.html
is a good place to start
a) To plot the two arbitrary points in three dimensions, follow these steps:
1. Choose the values of (x1, y1, z1) and (x2, y2, z2) for your points in 3D space. For example, let's say you choose (1, 2, 3) for the first point and (4, 5, 6) for the second point.
2. Draw a set of three perpendicular axes: the x-axis, y-axis, and z-axis. Each axis represents a different dimension. Make sure there are 90 degrees between the xy, xz, and yz planes.
3. Locate the first point (1, 2, 3) on the graph by finding the intersection of the three axes. Mark it as a point in the 3D space.
4. Similarly, locate the second point (4, 5, 6) on the graph by finding its intersection on the axes. Mark it as another point in the 3D space.
5. Connect the two points with a straight line segment, which represents the distance between them. Take note of the length of this line segment.
b) To generalize the equation for 3D distance, we can extend the Pythagorean Theorem to three dimensions. The distance formula would look as follows:
d = sqrt((x1 - x2)^2 + (y1 - y2)^2 + (z1 - z2)^2)
In this formula, (x1, y1, z1) represents the coordinates of the first point, and (x2, y2, z2) represents the coordinates of the second point.
To find the distance between these two points, plug the coordinates into the generalized formula and calculate the result.
For example, if we have the points (1, 2, 3) and (4, 5, 6), we can apply the formula:
d = sqrt((4 - 1)^2 + (5 - 2)^2 + (6 - 3)^2)
d = sqrt(3^2 + 3^2 + 3^2)
d = sqrt(9 + 9 + 9)
d = sqrt(27)
d ≈ 5.196
Therefore, the distance between the two points in this example is approximately 5.196 units.