# The points P=(5,2,-1), Q=(-3,-1,2), R=(21,8,-7) in \mathbb{R}^3 are three vertices of a...

## Question:

The points {eq}P=(5,2,-1), Q=(-3,-1,2), R=(21,8,-7) {/eq} in {eq}\mathbb{R}^3 {/eq}are three vertices of a parallelogram. Two of the sides in the parallelogram are given by the vectors {eq}\overrightarrow {PQ} {/eq} and {eq}\overrightarrow {PR}. {/eq} Find the coordinates of the fourth vertex, {eq}S {/eq}, of this parallelogram.

## Three-Dimensional Vector:

We know that the parallelogram has four sides and the opposite sides of the parallelogram will be parallel.

there is a property of the parallelogram that the diagonal of the parallelogram will bisect each other so the mid-point of the diagonal will be equal.

{eq}\text{Given}, P=(5,2,-1), Q=(-3,-1,2), R=(21,8,-7)\\ \text{Suppose,}\\ S = (x ,y ,z)\\ \text { we know that the diagonal of the parallelogram bisect each other so the mid-point of the 'QR' and 'PS' will be same. }\\ \displaystyle\frac{x+5}{2} =\displaystyle\frac {-3 + 21}{2}\\ x+5 = 18\\ x = 13\\ \displaystyle\frac{y+2}{2} =\displaystyle\frac {2+ 8}{2}\\ y+2 = 10 \\ y = 8\\ \displaystyle\frac{z-1}{2} =\displaystyle\frac {-1 - 7 }{2}\\ z - 1 = -8\\ z = -7\\ \text { So, } S = (13 , 8 ,-7) {/eq}