In contrast to the proof in Exercise 81, we now use coordinates and position vectors to prove the same result. Without loss of generality, let P(x₁, y₁, 0) and Q(x₂, y₂, 0) be two points in the xy-plane and let R(x₃, y₃, z₃) be a third point, such that P, Q, and R do not lie on a line. Consider ΔPQR.

a. Let M₁ be the midpoint of the side PQ. Find the coordinates of M₁ and the components of the vector

b. Find the vector from the origin to the point Z₁ two-thirds of the way along

c. Repeat the calculation of part (b) with the midpoint M₂ of RQ and the vectorto obtain the vector

d. Repeat the calculation of part (b) with the midpoint M₃ of PR and the vector to obtain the vector

e. Conclude that the medians of ΔPQR intersect at a point. Give the coordinates of the point.

f. With P(2, 4, 0), Q(4, 1, 0), and R(6, 3, 4), find the point at which the medians of ΔPQR intersect.

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