Thread: Proof of Midpoint and Vectors in 3 Dimensions

1. Proof of Midpoint and Vectors in 3 Dimensions

Let Q be the midpoint of PR and and T be the midpoint of SU of vectors, not coplanar in R^3 (3 dimensions). If the dot product of PT and TR = the dot product of SQ and QU = 0, show that the magnitude of PR is equal to the magnitude of SU.

I've been trying to rearrange the equations by using the distance formula and the dot product properties, like A dot B = magnitude of A * magnitude of B * cos(theta). Since the dot product of the new lines is 0, I know that these lines must make a 90 degree angle, but am still at a loss over how to show the lengths of PR and SU are the same.

I don't need a solution, just a way to get started because I am out of ideas.

2. Simple Geometry

Don't worry about the equations - draw a picture!

You have two right triangles, $\Delta PTR$ and $\Delta UQS$, where T bisects US and Q bisects PR.

Draw this and see if you can convince yourself that $\Delta TQR$ is congruent to $\Delta QTS$