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# Lesson 4: Lines, Planes, and the Distance Formula

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Using vectors and the various operations defined on them we can get equations for lines and planes based on descriptive data. We can also find distances between linear objects, such as point to line, point to plane, plane to plane, and line to line.

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### Lesson 4: Lines, Planes, and the Distance Formula

1. 1. Section 9.5 Equations of Lines and Planes Math 21a February 11, 2008 Announcements Oﬃce Hours Tuesday, Wednesday, 2–4pm (SC 323) All homework on the website No class Monday 2/18
2. 2. Outline Parallel and perpendicular in spaceland Lines in spaceland Lines in ﬂatland Equations for lines in spaceland Equations for planes Lines in ﬂatland, again Planes in spaceland Distances Point to line Point to plane line to line
3. 3. parallel and perpendicular quiz Determine whether each statement is true or false. 1. Two lines parallel to a third line are parallel. 2. Two lines perpendicular to a third line are parallel. 3. Two planes parallel to a third plane are parallel. 4. Two planes perpendicular to a third plane are parallel. 5. Two lines parallel to a plane are parallel. 6. Two lines perpendicular to a plane are parallel. 7. Two planes parallel to a line are parallel. 8. Two planes perpendicular to a line are parallel. 9. Two planes either intersect or are parallel. 10. Two lines either intersect or are parallel. 11. A plane and a line either intersect or are parallel.
4. 4. parallel and perpendicular quiz Determine whether each statement is true or false. 1. Two lines parallel to a third line are parallel. true 2. Two lines perpendicular to a third line are parallel. 3. Two planes parallel to a third plane are parallel. 4. Two planes perpendicular to a third plane are parallel. 5. Two lines parallel to a plane are parallel. 6. Two lines perpendicular to a plane are parallel. 7. Two planes parallel to a line are parallel. 8. Two planes perpendicular to a line are parallel. 9. Two planes either intersect or are parallel. 10. Two lines either intersect or are parallel. 11. A plane and a line either intersect or are parallel.
5. 5. parallel and perpendicular quiz Determine whether each statement is true or false. 1. Two lines parallel to a third line are parallel. true 2. Two lines perpendicular to a third line are parallel. false 3. Two planes parallel to a third plane are parallel. 4. Two planes perpendicular to a third plane are parallel. 5. Two lines parallel to a plane are parallel. 6. Two lines perpendicular to a plane are parallel. 7. Two planes parallel to a line are parallel. 8. Two planes perpendicular to a line are parallel. 9. Two planes either intersect or are parallel. 10. Two lines either intersect or are parallel. 11. A plane and a line either intersect or are parallel.
6. 6. parallel and perpendicular quiz Determine whether each statement is true or false. 1. Two lines parallel to a third line are parallel. true 2. Two lines perpendicular to a third line are parallel. false 3. Two planes parallel to a third plane are parallel. true 4. Two planes perpendicular to a third plane are parallel. 5. Two lines parallel to a plane are parallel. 6. Two lines perpendicular to a plane are parallel. 7. Two planes parallel to a line are parallel. 8. Two planes perpendicular to a line are parallel. 9. Two planes either intersect or are parallel. 10. Two lines either intersect or are parallel. 11. A plane and a line either intersect or are parallel.
7. 7. parallel and perpendicular quiz Determine whether each statement is true or false. 1. Two lines parallel to a third line are parallel. true 2. Two lines perpendicular to a third line are parallel. false 3. Two planes parallel to a third plane are parallel. true 4. Two planes perpendicular to a third plane are parallel. false 5. Two lines parallel to a plane are parallel. 6. Two lines perpendicular to a plane are parallel. 7. Two planes parallel to a line are parallel. 8. Two planes perpendicular to a line are parallel. 9. Two planes either intersect or are parallel. 10. Two lines either intersect or are parallel. 11. A plane and a line either intersect or are parallel.
8. 8. parallel and perpendicular quiz Determine whether each statement is true or false. 1. Two lines parallel to a third line are parallel. true 2. Two lines perpendicular to a third line are parallel. false 3. Two planes parallel to a third plane are parallel. true 4. Two planes perpendicular to a third plane are parallel. false 5. Two lines parallel to a plane are parallel. false 6. Two lines perpendicular to a plane are parallel. 7. Two planes parallel to a line are parallel. 8. Two planes perpendicular to a line are parallel. 9. Two planes either intersect or are parallel. 10. Two lines either intersect or are parallel. 11. A plane and a line either intersect or are parallel.
9. 9. parallel and perpendicular quiz Determine whether each statement is true or false. 1. Two lines parallel to a third line are parallel. true 2. Two lines perpendicular to a third line are parallel. false 3. Two planes parallel to a third plane are parallel. true 4. Two planes perpendicular to a third plane are parallel. false 5. Two lines parallel to a plane are parallel. false 6. Two lines perpendicular to a plane are parallel. true 7. Two planes parallel to a line are parallel. 8. Two planes perpendicular to a line are parallel. 9. Two planes either intersect or are parallel. 10. Two lines either intersect or are parallel. 11. A plane and a line either intersect or are parallel.
10. 10. parallel and perpendicular quiz Determine whether each statement is true or false. 1. Two lines parallel to a third line are parallel. true 2. Two lines perpendicular to a third line are parallel. false 3. Two planes parallel to a third plane are parallel. true 4. Two planes perpendicular to a third plane are parallel. false 5. Two lines parallel to a plane are parallel. false 6. Two lines perpendicular to a plane are parallel. true 7. Two planes parallel to a line are parallel. false 8. Two planes perpendicular to a line are parallel. 9. Two planes either intersect or are parallel. 10. Two lines either intersect or are parallel. 11. A plane and a line either intersect or are parallel.
11. 11. parallel and perpendicular quiz Determine whether each statement is true or false. 1. Two lines parallel to a third line are parallel. true 2. Two lines perpendicular to a third line are parallel. false 3. Two planes parallel to a third plane are parallel. true 4. Two planes perpendicular to a third plane are parallel. false 5. Two lines parallel to a plane are parallel. false 6. Two lines perpendicular to a plane are parallel. true 7. Two planes parallel to a line are parallel. false 8. Two planes perpendicular to a line are parallel. true 9. Two planes either intersect or are parallel. 10. Two lines either intersect or are parallel. 11. A plane and a line either intersect or are parallel.
12. 12. parallel and perpendicular quiz Determine whether each statement is true or false. 1. Two lines parallel to a third line are parallel. true 2. Two lines perpendicular to a third line are parallel. false 3. Two planes parallel to a third plane are parallel. true 4. Two planes perpendicular to a third plane are parallel. false 5. Two lines parallel to a plane are parallel. false 6. Two lines perpendicular to a plane are parallel. true 7. Two planes parallel to a line are parallel. false 8. Two planes perpendicular to a line are parallel. true 9. Two planes either intersect or are parallel. true 10. Two lines either intersect or are parallel. 11. A plane and a line either intersect or are parallel.
13. 13. parallel and perpendicular quiz Determine whether each statement is true or false. 1. Two lines parallel to a third line are parallel. true 2. Two lines perpendicular to a third line are parallel. false 3. Two planes parallel to a third plane are parallel. true 4. Two planes perpendicular to a third plane are parallel. false 5. Two lines parallel to a plane are parallel. false 6. Two lines perpendicular to a plane are parallel. true 7. Two planes parallel to a line are parallel. false 8. Two planes perpendicular to a line are parallel. true 9. Two planes either intersect or are parallel. true 10. Two lines either intersect or are parallel. false 11. A plane and a line either intersect or are parallel.
14. 14. parallel and perpendicular quiz Determine whether each statement is true or false. 1. Two lines parallel to a third line are parallel. true 2. Two lines perpendicular to a third line are parallel. false 3. Two planes parallel to a third plane are parallel. true 4. Two planes perpendicular to a third plane are parallel. false 5. Two lines parallel to a plane are parallel. false 6. Two lines perpendicular to a plane are parallel. true 7. Two planes parallel to a line are parallel. false 8. Two planes perpendicular to a line are parallel. true 9. Two planes either intersect or are parallel. true 10. Two lines either intersect or are parallel. false 11. A plane and a line either intersect or are parallel. true
15. 15. Parallelism in spaceland Two planes are parallel if they do not intersect A line and a plane are parallel if they do not intersect Two lines are skew if they are not both contained in a single plane Two lines are parallel if they are contained in a common plane and they do not intersect
16. 16. Outline Parallel and perpendicular in spaceland Lines in spaceland Lines in ﬂatland Equations for lines in spaceland Equations for planes Lines in ﬂatland, again Planes in spaceland Distances Point to line Point to plane line to line
17. 17. Lines in ﬂatland There are many ways to specify a line in the plane:
18. 18. Lines in ﬂatland There are many ways to specify a line in the plane: two points point and slope slope and intercept
19. 19. Lines in ﬂatland There are many ways to specify a line in the plane: two points point and slope slope and intercept How can we specify a line in three or more dimensions?
20. 20. Using vectors to describe lines Let y = mx + b be a line in the plane. Let
21. 21. Using vectors to describe lines Let y = mx + b be a line in the plane. Let r0 = 0, b r0
22. 22. Using vectors to describe lines Let y = mx + b be a line in the plane. Let v r0 = 0, b v = 1, m r0
23. 23. Using vectors to describe lines Let y = mx + b be a line in the plane. Let v r0 = 0, b v = 1, m r0 Then the line can be described as the set of all r(t) = r0 + tv as t ranges over all real numbers.
24. 24. Lines in spaceland Any line in R3 can be described by a point with position vector r0 and a direction vector v. It’s given by the vector equation r(t) = r0 + tv
25. 25. Lines in spaceland Any line in R3 can be described by a point with position vector r0 and a direction vector v. It’s given by the vector equation r(t) = r0 + tv If r = x0 , y0 , z0 and v = a, b, c , then the vector equation can be rewritten x, y , z = x0 + ta, y0 + tb, z0 + tc =⇒ x = x0 + at y = y0 + bt z = z0 + ct These are called the parametric equations for the line.
26. 26. Lines in spaceland Any line in R3 can be described by a point with position vector r0 and a direction vector v. It’s given by the vector equation r(t) = r0 + tv If r = x0 , y0 , z0 and v = a, b, c , then the vector equation can be rewritten x, y , z = x0 + ta, y0 + tb, z0 + tc =⇒ x = x0 + at y = y0 + bt z = z0 + ct These are called the parametric equations for the line. Solving the parametric equations for t gives x − x0 y − y0 z − z0 = = a b c These are called the symmetric equations for the line.
27. 27. Applying the deﬁnition Example Find the vector, parametric, and symmetric equations for the line that passes through (1, 2, 3) and (2, 3, 4).
28. 28. Applying the deﬁnition Example Find the vector, parametric, and symmetric equations for the line that passes through (1, 2, 3) and (2, 3, 4). Solution Use the initial vector 1, 2, 3 and direction vector 2, 3, 4 − 1, 2, 3 = 1, 1, 1 . Hence r(t) = 1, 2, 3 + t 1, 1, 1 The parametric equations are x =1+t y =2+t z =3+t The symmetric equations are x −1=y −2=z −3
29. 29. Another vector equation Alternatively, any line in R3 can be described by two points with position vectors r0 and r1 by letting r0 be the point and r1 − r0 the direction.
30. 30. Another vector equation Alternatively, any line in R3 can be described by two points with position vectors r0 and r1 by letting r0 be the point and r1 − r0 the direction. Then x = r0 + t(r1 − r0 ) = (1 − t)r0 + tr1 .
31. 31. Outline Parallel and perpendicular in spaceland Lines in spaceland Lines in ﬂatland Equations for lines in spaceland Equations for planes Lines in ﬂatland, again Planes in spaceland Distances Point to line Point to plane line to line
32. 32. Lines in ﬂatland, again Let n be perpendicular to v. n v r0
33. 33. Lines in ﬂatland, again r0 Let n be perpendicular to v. r− Then the head of r is on the n v line exactly when r − r0 is r parallel to v, or perpendicular r0 to n.
34. 34. Lines in ﬂatland, again r0 Let n be perpendicular to v. r− Then the head of r is on the n v line exactly when r − r0 is r parallel to v, or perpendicular r0 to n. So the line can be described as the set of all r such that n · (r − r0 ) = 0
35. 35. Generalizing again This generalizes to spaceland as well. z n r0 y x
36. 36. Generalizing again This generalizes to spaceland as well. z n r0 y x
37. 37. Generalizing again This generalizes to spaceland as well. z n r0 y x This time, the locus is a plane.
38. 38. Equations for planes The plane passing through the point with position vector r0 = x0 , y0 , z0 perpendicular to a, b, c has equations: The vector equation n · (r − r0 ) = 0 ⇐⇒ n · r = n · r0
39. 39. Equations for planes The plane passing through the point with position vector r0 = x0 , y0 , z0 perpendicular to a, b, c has equations: The vector equation n · (r − r0 ) = 0 ⇐⇒ n · r = n · r0 Rewriting the dot product in component terms gives the scalar equation a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0 The vector n is called a normal vector to the plane.
40. 40. Equations for planes The plane passing through the point with position vector r0 = x0 , y0 , z0 perpendicular to a, b, c has equations: The vector equation n · (r − r0 ) = 0 ⇐⇒ n · r = n · r0 Rewriting the dot product in component terms gives the scalar equation a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0 The vector n is called a normal vector to the plane. Rearranging this gives the linear equation ax + by + cz + d = 0, where d = −ax0 − by0 − cz0 .
41. 41. Example Find an equation of the plane that passes through the points P(1, 2, 3), Q(3, 5, 7), and R(4, 3, 1).
42. 42. Example Find an equation of the plane that passes through the points P(1, 2, 3), Q(3, 5, 7), and R(4, 3, 1). Solution −→ −→ − → Let r0 = OP = 1, 2, 3 . To get n, take PQ × PR: i j k −→ − → PQ × PR = 2 3 4 = −10, 16, −7 3 1 −2 So the scalar equation is −10(x − 1) + 16(y − 2) − 7(z − 3) = 0.
43. 43. Outline Parallel and perpendicular in spaceland Lines in spaceland Lines in ﬂatland Equations for lines in spaceland Equations for planes Lines in ﬂatland, again Planes in spaceland Distances Point to line Point to plane line to line
44. 44. Distance from point to line Deﬁnition The distance between a point and a line is the smallest distance from that point to all points on the line. You can ﬁnd it by projection. Q v θ P0
45. 45. Distance from point to line Deﬁnition The distance between a point and a line is the smallest distance from that point to all points on the line. You can ﬁnd it by projection. Q b v θ P0
46. 46. Distance from point to line Deﬁnition The distance between a point and a line is the smallest distance from that point to all points on the line. You can ﬁnd it by projection. Q b b·v projv b = v v v·v θ P0
47. 47. Distance from point to line Deﬁnition The distance between a point and a line is the smallest distance from that point to all points on the line. You can ﬁnd it by projection. Q b·v b− v v·v b b·v projv b = v v v·v θ P0
48. 48. Example Find the distance between the point (4, 6) and the line x − 2y + 3 = 0.
49. 49. Example Find the distance between the point (4, 6) and the line x − 2y + 3 = 0. Solution The line goes through (1, 2) and has slope 1/2, so we can use v = 2, 1 and b = 3, 4 . Then the projection of b on the line is given by b·v 10 projv b = v= 2, 1 = 4, 2 v·v 5 So b − projv b = 3, 4 − 4, 2 = −1, 2 (Notice that 2, 1 and −1, 2 are perpendicular.) So the distance is √ | −1, 2 | = 5
50. 50. Point to plane Deﬁnition The distance between a point and a plane is the smallest distance from that point to all points on the line. Q b |n · b| |n| n P0 To ﬁnd the distance from the a point to a plane, project the displacement vector from any point on the plane to the given point onto the normal vector.
51. 51. We have |n · b| D= |n| If Q = (x1 , y1 , z1 ), and the plane is given by ax + by + cz + d = 0, then n = a, b, c , and n · b = a, b, c · x1 − x0 , y1 − y0 , z1 − z0 = ax1 + by1 + cz1 − ax0 − by0 − cz0 = ax1 + by1 + cz1 + d
52. 52. We have |n · b| D= |n| If Q = (x1 , y1 , z1 ), and the plane is given by ax + by + cz + d = 0, then n = a, b, c , and n · b = a, b, c · x1 − x0 , y1 − y0 , z1 − z0 = ax1 + by1 + cz1 − ax0 − by0 − cz0 = ax1 + by1 + cz1 + d So the distance between the plane ax + by + cz + d = 0 and the point (x1 , y1 , z1 ) is |ax1 + by1 + cz1 + d| D= √ a2 + b 2 + c 2
53. 53. Example Find the distance between the plane containing the three points P(1, 2, 3), Q(3, 5, 7), and R(4, 3, 1) and the origin.
54. 54. Example Find the distance between the plane containing the three points P(1, 2, 3), Q(3, 5, 7), and R(4, 3, 1) and the origin. Solution We’ve already found the plane has scalar equation given by 0 = −10(x − 1) + 16(y − 2) − 7(z − 3) = −10x + 16y − 7z − 1 So d = 1. Using the formula above with (x1 , y1 , z1 ) = (0, 0, 0) we have 1 1 D=√ = √ 102 + 162 + 72 9 5
55. 55. line to line To ﬁnd the distance between two skew lines, create two parallel planes and ﬁnd the distance between a point in one to the other. For an example, see Example 10 on page 673.