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

1 Answer
Aug 14, 2014

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.