The amazing quadrilateral property—coordinate free The points P , Q , R , and S , joined by the vectors u , v , w , and x , are the vertices of a quadrilateral in ℝ 3 . The four points needn’t lie in a plane (see figure). Use the following steps to prove that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. The proof does not use a coordinate system . a. Use vector addition to show that u + v = w + x . b. Let m be the vector that joins the midpoints of PQ and QR. Show that m = ( u + v )/2. c. Let n be the vector that joins the midpoints of PS and SR. Show that n = ( x + w )/2. d. Combine parts (a), (b), and (c) to conclude that m = n . e. Explain why part (d) implies that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram.
The amazing quadrilateral property—coordinate free The points P , Q , R , and S , joined by the vectors u , v , w , and x , are the vertices of a quadrilateral in ℝ 3 . The four points needn’t lie in a plane (see figure). Use the following steps to prove that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. The proof does not use a coordinate system . a. Use vector addition to show that u + v = w + x . b. Let m be the vector that joins the midpoints of PQ and QR. Show that m = ( u + v )/2. c. Let n be the vector that joins the midpoints of PS and SR. Show that n = ( x + w )/2. d. Combine parts (a), (b), and (c) to conclude that m = n . e. Explain why part (d) implies that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram.
Solution Summary: The author explains how the value of u+v=w+x is obtained by vector addition.
The amazing quadrilateral property—coordinate free The points P, Q, R, and S, joined by the vectorsu, v, w, and x, are the vertices of a quadrilateral in
ℝ
3
. The four points needn’t lie in a plane (see figure). Use the following steps to prove that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. The proof does not use a coordinate system.
a. Use vector addition to show that u + v = w + x.
b. Let m be the vector that joins the midpoints of PQ and QR. Show that m = (u + v)/2.
c. Let n be the vector that joins the midpoints of PS and SR. Show that n = (x + w)/2.
d. Combine parts (a), (b), and (c) to conclude that m = n.
e. Explain why part (d) implies that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram.
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
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