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Mathematics 13: Lecture 4Planes
Dan Sloughter
Furman University
January 10, 2008
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 1 / 10
Planes in Rn
I Suppose ~v and ~w are nonzero vectors in Rn which are not parallel.Given any third vector ~p0 in Rn, we call the set of all points P
~p = ~p0 + s~v + t~w ,
where s and t are scalars, a plane.
I We call this equation a vector equation for P.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 2 / 10
Planes in Rn
I Suppose ~v and ~w are nonzero vectors in Rn which are not parallel.Given any third vector ~p0 in Rn, we call the set of all points P
~p = ~p0 + s~v + t~w ,
where s and t are scalars, a plane.
I We call this equation a vector equation for P.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 2 / 10
Parametric equations
I If
~p =
x1
x2...xn
, ~p0 =
p1
p2...
pn
, ~v =
v1
v2...vn
, and ~w =
w1
w2...
wn
,
then
x1 = p1 + sv1 + tw1,
x2 = p2 + sv2 + tw2,
... =...
xn = pn + svn + twn,
which are the scalar, or parametric, equations for P.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 3 / 10
Example
I Let P be the plane in R3 which contains the points
~p0 =
1−1
1
, ~q =
122
, and ~r =
−113
.
I Then
~v = ~q − ~p0 =
031
and ~w = ~r − ~p0 =
−222
are vectors which lie on P.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 4 / 10
Example
I Let P be the plane in R3 which contains the points
~p0 =
1−1
1
, ~q =
122
, and ~r =
−113
.
I Then
~v = ~q − ~p0 =
031
and ~w = ~r − ~p0 =
−222
are vectors which lie on P.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 4 / 10
Example (cont’d)
I So the vector equation of P is
~p =
1−1
1
+ s
031
+ t
−222
,
and the parametric equations are
x = 1− 2t,
y = −1 + 3s + 2t,
z = 1 + s + 2t.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 5 / 10
Normal form in R3
I Suppose P is a plane in R3 with vector equation ~p = ~p0 + s~v + t~w .
I Analogous to the case with lines in R2, if ~n is perpendicular to both ~vand ~w , then we may describe P as the set of all ~p in R3 satisfying~n · (~p − ~p0) = 0.
I This is the normal equation of P.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 6 / 10
Normal form in R3
I Suppose P is a plane in R3 with vector equation ~p = ~p0 + s~v + t~w .
I Analogous to the case with lines in R2, if ~n is perpendicular to both ~vand ~w , then we may describe P as the set of all ~p in R3 satisfying~n · (~p − ~p0) = 0.
I This is the normal equation of P.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 6 / 10
Normal form in R3
I Suppose P is a plane in R3 with vector equation ~p = ~p0 + s~v + t~w .
I Analogous to the case with lines in R2, if ~n is perpendicular to both ~vand ~w , then we may describe P as the set of all ~p in R3 satisfying~n · (~p − ~p0) = 0.
I This is the normal equation of P.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 6 / 10
ExampleI Suppose P is a plane passing through
~p0 =
1−1
1
with normal vector
~n =
4−2
6
.
I The normal form of the equation for P is 4−2
6
·x
yz
− 1−1
1
= 0.
I Or, equivalently,4x − 2y + 6z = 12.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 7 / 10
ExampleI Suppose P is a plane passing through
~p0 =
1−1
1
with normal vector
~n =
4−2
6
.
I The normal form of the equation for P is 4−2
6
·x
yz
− 1−1
1
= 0.
I Or, equivalently,4x − 2y + 6z = 12.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 7 / 10
ExampleI Suppose P is a plane passing through
~p0 =
1−1
1
with normal vector
~n =
4−2
6
.
I The normal form of the equation for P is 4−2
6
·x
yz
− 1−1
1
= 0.
I Or, equivalently,4x − 2y + 6z = 12.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 7 / 10
Distance to a plane
I Let P be a plane in R3 with normal equation ~n · (~p − ~p0) = 0. Let Dbe the shortest distance from a point ~q to P.
I Let ~v = ~q − ~p0.
I Note: the component of ~v in the direction of ~n is
~n · ~v‖~n‖
=~n · ~q − ~n · ~p0
‖~n‖.
I Hence
D =|~n · ~q − ~n · ~p0|
‖~n‖.
I Note: the point on P closest to ~q is ~q − proj~n~v .
I Note: an analogous formula works for the distance from a point to aline in R2.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 8 / 10
Distance to a plane
I Let P be a plane in R3 with normal equation ~n · (~p − ~p0) = 0. Let Dbe the shortest distance from a point ~q to P.
I Let ~v = ~q − ~p0.
I Note: the component of ~v in the direction of ~n is
~n · ~v‖~n‖
=~n · ~q − ~n · ~p0
‖~n‖.
I Hence
D =|~n · ~q − ~n · ~p0|
‖~n‖.
I Note: the point on P closest to ~q is ~q − proj~n~v .
I Note: an analogous formula works for the distance from a point to aline in R2.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 8 / 10
Distance to a plane
I Let P be a plane in R3 with normal equation ~n · (~p − ~p0) = 0. Let Dbe the shortest distance from a point ~q to P.
I Let ~v = ~q − ~p0.
I Note: the component of ~v in the direction of ~n is
~n · ~v‖~n‖
=~n · ~q − ~n · ~p0
‖~n‖.
I Hence
D =|~n · ~q − ~n · ~p0|
‖~n‖.
I Note: the point on P closest to ~q is ~q − proj~n~v .
I Note: an analogous formula works for the distance from a point to aline in R2.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 8 / 10
Distance to a plane
I Let P be a plane in R3 with normal equation ~n · (~p − ~p0) = 0. Let Dbe the shortest distance from a point ~q to P.
I Let ~v = ~q − ~p0.
I Note: the component of ~v in the direction of ~n is
~n · ~v‖~n‖
=~n · ~q − ~n · ~p0
‖~n‖.
I Hence
D =|~n · ~q − ~n · ~p0|
‖~n‖.
I Note: the point on P closest to ~q is ~q − proj~n~v .
I Note: an analogous formula works for the distance from a point to aline in R2.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 8 / 10
Distance to a plane
I Let P be a plane in R3 with normal equation ~n · (~p − ~p0) = 0. Let Dbe the shortest distance from a point ~q to P.
I Let ~v = ~q − ~p0.
I Note: the component of ~v in the direction of ~n is
~n · ~v‖~n‖
=~n · ~q − ~n · ~p0
‖~n‖.
I Hence
D =|~n · ~q − ~n · ~p0|
‖~n‖.
I Note: the point on P closest to ~q is ~q − proj~n~v .
I Note: an analogous formula works for the distance from a point to aline in R2.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 8 / 10
Distance to a plane
I Let P be a plane in R3 with normal equation ~n · (~p − ~p0) = 0. Let Dbe the shortest distance from a point ~q to P.
I Let ~v = ~q − ~p0.
I Note: the component of ~v in the direction of ~n is
~n · ~v‖~n‖
=~n · ~q − ~n · ~p0
‖~n‖.
I Hence
D =|~n · ~q − ~n · ~p0|
‖~n‖.
I Note: the point on P closest to ~q is ~q − proj~n~v .
I Note: an analogous formula works for the distance from a point to aline in R2.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 8 / 10
Example
I Suppose P is a plane with equation 4x + 2y + 4z = 8 and
~q =
334
.
I Note:
~n =
424
is a normal vector for P.
I Moreover, for any point ~p0 on P, ~n · ~p0 = 8.
I Hence the shortest distance D from ~q to P is
D =|~n · ~q − 8|‖~n‖
=|34− 8|
6=
13
3.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 9 / 10
Example
I Suppose P is a plane with equation 4x + 2y + 4z = 8 and
~q =
334
.
I Note:
~n =
424
is a normal vector for P.
I Moreover, for any point ~p0 on P, ~n · ~p0 = 8.
I Hence the shortest distance D from ~q to P is
D =|~n · ~q − 8|‖~n‖
=|34− 8|
6=
13
3.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 9 / 10
Example
I Suppose P is a plane with equation 4x + 2y + 4z = 8 and
~q =
334
.
I Note:
~n =
424
is a normal vector for P.
I Moreover, for any point ~p0 on P, ~n · ~p0 = 8.
I Hence the shortest distance D from ~q to P is
D =|~n · ~q − 8|‖~n‖
=|34− 8|
6=
13
3.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 9 / 10
Example
I Suppose P is a plane with equation 4x + 2y + 4z = 8 and
~q =
334
.
I Note:
~n =
424
is a normal vector for P.
I Moreover, for any point ~p0 on P, ~n · ~p0 = 8.
I Hence the shortest distance D from ~q to P is
D =|~n · ~q − 8|‖~n‖
=|34− 8|
6=
13
3.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 9 / 10
Example (cont’d)
I Note: If ~v = ~q − ~p0, where ~p0 is some point on P, then
proj~n~v =26
36
424
=13
9
212
.
I So the point on P closest to ~q is
~q − proj~n~v =
19149109
.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 10 / 10
Example (cont’d)
I Note: If ~v = ~q − ~p0, where ~p0 is some point on P, then
proj~n~v =26
36
424
=13
9
212
.
I So the point on P closest to ~q is
~q − proj~n~v =
19149109
.
Dan Sloughter (Furman University) Mathematics 13: Lecture 4 January 10, 2008 10 / 10