Question

What is the distance between `M(7,−1,5)` and the origin?

Answer 1

The answer is `5sqrt(3)`.

We simply use the Pythagorean Theorem for this calculation. We can draw a right triangle on the `xy` plane and extend this for another right triangle to the `z` plane. Let `d` be the length of the diagonal on the `xy` plane. Then

`l=sqrt(d^2+z^2)`

but

`d^2=x^2+y^2`

so

`l=sqrt(x^2+y^2+z^2)`

and substituting the values from your question:

`l=sqrt(7^2+(-1)^2+5^2)=sqrt(49+1+25)=sqrt(75)=5sqrt(3)`

We can generalize the distance between any 2 - 3D points as:

`l=sqrt((P_(2_x)-P_(1_x))^2+(P_(2_y)-P_(1_y))^2+(P_(2_z)-P_(1_z))^2)`

The order of `P_1` and `P_2` doesn't matter because the difference is squared which always results in a positive value.