The points P, Q, R, and S, joined by the vectors u, v, w, x, are the vertices of a quadrilateral in R3. 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. Use vector addition to show that u + v = w + x. Choose the correct answer below. u + v = w + x Let m be the vector that joints the midpoints of PQ and QR. Show that m = (u + v)/2. Choose the correct answer below. Let n be the vector that joints the midpoints of PS and SR. Show that n = (x + w)/2. Choose the correct answer below. Combine parts (a), (b), and (c) to conclude that m = n. Since both u + v and w + x are equal to , m and n are both equal to , and so m = n. Since both u + v and w + x are equal to , m and n are both equal to 1/2 , and so m = n. Since both u + v and w + x are equal to , m and n are both equal to , and so m = n. Since both u + v and w + x are equal to , m and n are both equal to 1/2 , and so m = n. Explain why part (d) implies that the lines segments joining the midpoints of the sides of the quadrilateral form a parallelogram. Using a similar process, it can be shown that the side joining the tails and the side joining the heads of m and n (denoted by dashed lines in the figure) also have the same length and direction, forming a parallelogram. Since m = n, u and w are parallel, and x and v are parallel. It follows that each pair of parallel vectors have equal length, forming a parallelogram. Since m and n have the same direction, they are parallel. Since m and n are parallel and have the same length, the side joining the tails and the side joining the heads of m n (denoted by dashed lines in the figure) must also be of equal length and be parallel, forming a parallelogram.
Show transcribed image text The points P, Q, R, and S, joined by the vectors u, v, w, x, are the vertices of a quadrilateral in R3. 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. Use vector addition to show that u + v = w + x. Choose the correct answer below. u + v = w + x Let m be the vector that joints the midpoints of PQ and QR. Show that m = (u + v)/2. Choose the correct answer below. Let n be the vector that joints the midpoints of PS and SR. Show that n = (x + w)/2. Choose the correct answer below. Combine parts (a), (b), and (c) to conclude that m = n. Since both u + v and w + x are equal to , m and n are both equal to , and so m = n. Since both u + v and w + x are equal to , m and n are both equal to 1/2 , and so m = n. Since both u + v and w + x are equal to , m and n are both equal to , and so m = n. Since both u + v and w + x are equal to , m and n are both equal to 1/2 , and so m = n. Explain why part (d) implies that the lines segments joining the midpoints of the sides of the quadrilateral form a parallelogram. Using a similar process, it can be shown that the side joining the tails and the side joining the heads of m and n (denoted by dashed lines in the figure) also have the same length and direction, forming a parallelogram. Since m = n, u and w are parallel, and x and v are parallel. It follows that each pair of parallel vectors have equal length, forming a parallelogram. Since m and n have the same direction, they are parallel. Since m and n are parallel and have the same length, the side joining the tails and the side joining the heads of m n (denoted by dashed lines in the figure) must also be of equal length and be parallel, forming a parallelogram.