Mathematics 13: Lecture Planes Dan Sloughter Furman University January 10, 2008 Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, 2008 1 / 10
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, 2008 2 / 10
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, 2008 2 / 10
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, 2008 3 / 10
Example Let P be the plane in R 3 which contains the points 1 1 1 p 0 = 1, q = 2, and r = 1. 1 2 3 Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, 2008 / 10
Example Let P be the plane in R 3 which contains the points 1 1 1 p 0 = 1, q = 2, and r = 1. 1 2 3 Then 0 2 v = q p 0 = 3 and w = r p 0 = 1 2 2 are vectors which lie on P. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, 2008 / 10
Example (cont d) So the vector equation of P is 1 0 2 p = 1 + s 3 + t 2, 1 1 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, 2008 5 / 10
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, 2008 6 / 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. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, 2008 6 / 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, 2008 6 / 10
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, 2008 7 / 10
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, 2008 7 / 10
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, 2008 7 / 10
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, 2008 8 / 10
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, 2008 8 / 10
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, 2008 8 / 10
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, 2008 8 / 10
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, 2008 8 / 10
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, 2008 8 / 10
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, 2008 9 / 10
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, 2008 9 / 10
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, 2008 9 / 10
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 = 3 8 6 = 13 3. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, 2008 9 / 10
Example (cont d) Note: If v = q p 0, where p 0 is some point on P, then proj n v = 26 2 = 13 2 1. 36 9 2 Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, 2008 10 / 10
Example (cont d) Note: If v = q p 0, where p 0 is some point on P, then proj n v = 26 2 = 13 2 1. 36 9 2 So the point on P closest to q is q proj n v = 1 9 1 9 10 9. Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, 2008 10 / 10