Math 20C. Lecture Examples.

Transcription

1 Math 20C. Lecture Eamples. (8/1/08) Section Vectors in three dimensions To use rectangular z-coordinates in three-dimensional space, e introduce mutuall perpendicular -, -, and z-aes intersecting at their origins, as in Figure 1. These aes form -, z-, and z-coordinate planes, hich diide the space into eight octants. FIGURE 1 FIGURE 2 The coordinates (,, z) of a point in space are determined b planes passing through the point and perpendicular to the coordinate aes (Figure 2). Eample 1 Sketch the bo consisting of all points (,,z) ith 0 2, 2 3, and 0 z 2. What are the coordinates of the eight corners of the bo? Anser: Figure A1. The corners of its base, ordered counterclockise, are (2, 2, 0), (2, 3, 0), (0, 3, 0), and (0, 2, 0). The corners of its top are (2, 2, 2), (2, 3, 2), (0, 3, 2), and (0, 2, 2). Figure A1 Lecture notes to accompan Section 12.2 of Calculus, Earl Transcendentals b Rogaski. 1

2 Math 20C. Lecture Eamples. (8/1/08) Section 12.2, p. 2 The Pthagorean Theorem and the distance beteen to points If a rectangular bo has length a, idth b, and height c, as in Figure 6, then, b the Pthagorean Theorem for a right triangle, the length of a diagonal of its base is a 2 + b 2. Then, because the diagonal of the bo is the hpotenuse of a right triangle ith base of length a 2 + b 2 and height c (Figure 7), its length is the square root of [ a 2 + b 2 ] 2 + c 2 = a 2 + b 2 + c 2. This gies the Pthagorean Theorem in space: [ The length of a diagonal of a rectangular bo ith sides a,b, and c is ] = a 2 + b 2 + c 2. FIGURE 6 FIGURE 7 Eample 2 What is the length of the diagonals of the bo from Eample 1? Anser: The length of each of its four diagonals is = 3 Because points P = ( 1, 1, z 1 ) and Q = ( 2, 2, z 2 ) in z-space are at diagonall opposite corners of a rectangular bo ith sides of lengths 2 1, 2 1, and z 2 z 1, the distance PQ beteen the points is PQ = ( 2 1 ) 2 + ( 2 1 ) 2 + (z 2 z 1 ) 2. Eample 3 Describe the set of points defined b the equation ( 1) 2 + ( 2) 2 + (z 3) 2 = 16. Anser: ( 1) 2 + ( 2) 2 + (z 3) 2 = 16 is the sphere of radius 4 ith its center at (1, 2, 3). Eample 4 Describe the set of points defined b the equation = 25. Anser: = 25 is the clinder of radius 5 ith the z-ais as its ais. Vectors in space A nonzero ector in z- space, like a nonzero ector in an -plane, represents a positie number and a direction. If e put the base of the ector at the origin, as in Figure 8, then the coordinates (a,b, c) of its tip are the -, -, and z-components of the ector and e rite = a,b, c. The zero ector 0 = 0, 0, 0 has zero length and no direction. FIGURE 8

3 Section 12.2, p. 3 Math 20C. Lecture Eamples. (8/1/08) The Pthagorean Theorem in space shos that the length of the ector = a,b, c is equal to the square root of the sum of the squares of its components: c = a, b, c = a 2 + b 2 + c 2. The rules for adding to ectors in space and multipling a ector in space b a real number are analogous to those for ectors in a plane: Definition 1 For an ectors = a 1, b 1, c 1 and = a 2,b 2,c 2 and an number λ, + = a 1,b 1, c 1 + a 2, b 2,c 2 = a 1 + a 2, b 1 + b 2, c 1 + c 2 λ = λ a 1, b 1, c 1 = λa 1,λb 1,λc 1. These operations and the subtraction of ectors in space hae the same geometric interpretations as in an -plane (see Figures 9 through 12.) + d + b a c FIGURE 9 FIGURE 10 λ (λ > 0) µ (µ < 0) FIGURE 11 FIGURE 12

4 Math 20C. Lecture Eamples. (8/1/08) Section 12.2, p. 4 The unit ectors i,j, and k In the last section e epressed the ector a,b in the plane as ai + bj here i and j are unit ectors in the directions of the positie - and -aes, respectiel. In three dimensions, e also use a third unit ector k in the direction of the positie z-ais, as in Figure 13. Then a,b, c = ai + bj + ck for an a,b, and c (Figure 14). FIGURE 13 FIGURE 14 Eample 5 Write z = u in the form ai + bj + ck, here u = 3i j, = j 3k and = i + k. Anser: z = 6i + j 3k The position ector OP of a point (,,z) in space is,,z (Figure 15). The displacement ector PQ from P = ( 1, 1, z 1 ) to Q = ( 2, 2, z 2 ) is PQ = 2 1, 2 1, z 2 z 1 as is shon in Figure 16 for a case here 2 1 and z 2 z 1 are positie and 2 1 is negatie. FIGURE 15 FIGURE 16 PQ = 2 1, 2 1,z 2 z 1

5 Section 12.2, p. 5 Math 20C. Lecture Eamples. (8/1/08) Eample 6 Three adjacent ertices of a parallelogram PQRS in space are P = (1, 3, 2), Q = (4, 5, 3), and R = (2, 1, 0). What are the coordinates of the point S opposite Q? Anser: Use the schematic sketch in Figure A6. S = ( 1, 3, 1) Figure A6 Parametric equations of lines in space A line in z-space can be described b giing the coordinates of a point P = ( 0, 0, z 0 ) on it and a nonzero ector = a, b, c parallel to it, as in Figure 17. P OP FIGURE 17 O Theorem 1 The line L through the point P = ( 0, 0, z 0 ) and parallel to the nonzero ector = a,b, c in z-space has the parametric equations, = 0 + at L: = 0 + bt z = z 0 + ct. Eample 7 Eample 8 Eample 9 Gie parametric equations for the line L through the point (6,4, 3) and parallel to the ector 2i + 5j 7k. Anser: L: = 6 + 2t, = 4 + 5t, z = 3 7t Gie parametric equations for the line L through P = (5, 3, 1) and Q = (7, 2,0). Anser: L: = 5 + 2t, = 3 5t, z = 1 t Find the intersection of the lines L 1 : = 2 t, = 3 + t,z = 4 2t and L 2 : = 3 + t, = 1 + 2t,z = 9 3t Anser: Intersection: (0, 5, 0) Interactie Eamples Work the folloing Interactie Eamples on Shenk s eb page, http//.math.ucsd.edu/ ashenk/: Section 12.3: Eamples 1, 2, and 6 Section 12.5: Eamples 1 and 2 The chapter and section numbers on Shenk s eb site refer to his calculus manuscript and not to the chapters and sections of the tetbook for the course.