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Does it seem obvious that the midpoint between two points, say P and Q, doesn t depend on whether we go from P to Q or from Q to P That s indeed the case; but if we demand proof, we must show that for real numbers xp, yp, zp, xq, yq, and zq, it s always true that [(xp + xq)/2,(yp + yq)/2,(zp + zq)/2] = [(xq + xp)/2,(yq + yp)/2,(zq + zp)/2] This proof is almost trivial, but it s good mental exercise to put it down in rigorous form The commutative law for addition of real numbers tells us that xp + xq = xq + xp Dividing each side by 2 gives us (xp + xq)/2 = (xq + xp)/2 Using the same logic with the y and z coordinates, we get (yp + yq)/2 = (yq + yp)/2 and (zp + zq)/2 = (zq + zp)/2 Based on these facts, we know that the coordinates on both sides of the original equation are identical It follows that the midpoint along a straight-line segment connecting any two points in Cartesian three-space is the same, regardless of which way we go
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Imagine two points in Cartesian three-space where corresponding coordinates are negatives of each other Show that the midpoint is exactly at the origin
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We can choose any point P whose coordinates are all real numbers Let s suppose that P = (xp,yp,zp) Then the coordinates of Q are Q = ( xp, yp, zp) The coordinates of the midpoint M are (xm,ym,zm) = {[(xp + ( xp)]/2,[(yp + ( yp)]/2,[(zp + ( zp)/2]} = [(xp xp)/2,(yp yp)/2,(zp zp)/2] = (0/2,0/2,0/2) = (0,0,0) The point (0,0,0) is, of course, the origin of the coordinate system
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This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 What are the individual x, y, and z coordinates of the three points P, Q, and R shown in Fig 7-7
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+y Q ( 5, 4, 0) Origin = (0, 0, 0) L x R (0, 0, 6) Each axis increment is 1 unit N
M y
P (3, 3, 4)
Figure 7-7 Illustration for Problems 1 through 10 Each axis division
represents 1 unit
2 Determine the distance of the point P from the origin in Fig 7-7 Using a calculator, approximate the answer by rounding off to three decimal places 3 Determine the distance of the point Q from the origin in Fig 7-7 Using a calculator, approximate the answer by rounding off to three decimal places 4 Determine the distance of the point R from the origin in Fig 7-7 This should come out exact, so you won t need a calculator! 5 Determine the length of the line segment L in Fig 7-7 Using a calculator, approximate the answer by rounding off to three decimal places 6 Determine the length of the line segment M in Fig 7-7 Using a calculator, approximate the answer by rounding off to three decimal places
Practice Exercises
7 Determine the length of the line segment N in Fig 7-7 Using a calculator, approximate the answer by rounding off to three decimal places 8 Determine the coordinates of the midpoint of line segment L in Fig 7-7 9 Determine the coordinates of the midpoint of line segment M in Fig 7-7 10 Determine the coordinates of the midpoint of line segment N in Fig 7-7
CHAPTER
Vectors in Cartesian Three-Space
We ve learned how to work with Cartesian coordinates in two and three dimensions, and we ve learned about vectors in two dimensions Now it s time to explore how vectors behave in Cartesian xyz space