Mathematics 13: Lecture 4

Transcription

1 Mathematics 13: Lecture Planes Dan Sloughter Furman University January 10, 2008 Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

2 Planes in R n Suppose v and w are nonzero vectors in R n which are not parallel. Given any third vector p 0 in R n, we call the set of all points P where s and t are scalars, a plane. p = p 0 + s v + t w, Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

3 Planes in R n Suppose v and w are nonzero vectors in R n which are not parallel. Given any third vector p 0 in R n, we call the set of all points P where s and t are scalars, a plane. p = p 0 + s v + t w, We call this equation a vector equation for P. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

4 Parametric equations If then x 1 p 1 x 2 p =., p p 2 0 =., v = v 2 w 2, and w =.., x n p n v n w n v 1 w 1 x 1 = p 1 + sv 1 + tw 1, x 2 = p 2 + sv 2 + tw 2,. =. x n = p n + sv n + tw n, which are the scalar, or parametric, equations for P. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

5 Example Let P be the plane in R 3 which contains the points p 0 = 1, q = 2, and r = Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, 2008 / 10

6 Example Let P be the plane in R 3 which contains the points p 0 = 1, q = 2, and r = Then 0 2 v = q p 0 = 3 and w = r p 0 = are vectors which lie on P. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, 2008 / 10

7 Example (cont d) So the vector equation of P is p = 1 + s 3 + t 2, and the parametric equations are x = 1 2t, y = 1 + 3s + 2t, z = 1 + s + 2t. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

8 Normal form in R 3 Suppose P is a plane in R 3 with vector equation p = p 0 + s v + t w. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

9 Normal form in R 3 Suppose P is a plane in R 3 with vector equation p = p 0 + s v + t w. Analogous to the case with lines in R 2, if n is perpendicular to both v and w, then we may describe P as the set of all p in R 3 satisfying n ( p p 0 ) = 0. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

10 Normal form in R 3 Suppose P is a plane in R 3 with vector equation p = p 0 + s v + t w. Analogous to the case with lines in R 2, if n is perpendicular to both v and w, then we may describe P as the set of all p in R 3 satisfying n ( p p 0 ) = 0. This is the normal equation of P. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

11 Example Suppose P is a plane passing through 1 p 0 = 1 1 with normal vector n = 2. 6 Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

12 Example Suppose P is a plane passing through 1 p 0 = 1 1 with normal vector n = 2. 6 The normal form of the equation for P is x 1 2 y 1 = 0. 6 z 1 Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

13 Example Suppose P is a plane passing through 1 p 0 = 1 1 with normal vector n = 2. 6 The normal form of the equation for P is x 1 2 y 1 = 0. 6 z 1 Or, equivalently, x 2y + 6z = 12. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

14 Distance to a plane Let P be a plane in R 3 with normal equation n ( p p 0 ) = 0. Let D be the shortest distance from a point q to P. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

15 Distance to a plane Let P be a plane in R 3 with normal equation n ( p p 0 ) = 0. Let D be the shortest distance from a point q to P. Let v = q p 0. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

16 Distance to a plane Let P be a plane in R 3 with normal equation n ( p p 0 ) = 0. Let D be the shortest distance from a point q to P. Let v = q p 0. Note: the component of v in the direction of n is n v n = n q n p 0. n Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

17 Distance to a plane Let P be a plane in R 3 with normal equation n ( p p 0 ) = 0. Let D be the shortest distance from a point q to P. Let v = q p 0. Note: the component of v in the direction of n is n v n = n q n p 0. n Hence D = n q n p 0. n Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

18 Distance to a plane Let P be a plane in R 3 with normal equation n ( p p 0 ) = 0. Let D be the shortest distance from a point q to P. Let v = q p 0. Note: the component of v in the direction of n is n v n = n q n p 0. n Hence D = n q n p 0. n Note: the point on P closest to q is q proj n v. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

19 Distance to a plane Let P be a plane in R 3 with normal equation n ( p p 0 ) = 0. Let D be the shortest distance from a point q to P. Let v = q p 0. Note: the component of v in the direction of n is n v n = n q n p 0. n Hence D = n q n p 0. n Note: the point on P closest to q is q proj n v. Note: an analogous formula works for the distance from a point to a line in R 2. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

20 Example Suppose P is a plane with equation x + 2y + z = 8 and 3 q = 3. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

21 Example Suppose P is a plane with equation x + 2y + z = 8 and 3 q = 3. Note: is a normal vector for P. n = 2 Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

22 Example Suppose P is a plane with equation x + 2y + z = 8 and 3 q = 3. Note: is a normal vector for P. n = 2 Moreover, for any point p 0 on P, n p 0 = 8. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

23 Example Suppose P is a plane with equation x + 2y + z = 8 and 3 q = 3. Note: is a normal vector for P. n = 2 Moreover, for any point p 0 on P, n p 0 = 8. Hence the shortest distance D from q to P is D = n q 8 n = = Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

24 Example (cont d) Note: If v = q p 0, where p 0 is some point on P, then proj n v = 26 2 = Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10

25 Example (cont d) Note: If v = q p 0, where p 0 is some point on P, then proj n v = 26 2 = So the point on P closest to q is q proj n v = Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, / 10