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Lines and Planes in SpaceMATH 311, Calculus III
J. Robert Buchanan
Department of Mathematics
Summer 2010
J. Robert Buchanan Lines and Planes in Space
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Lines and Points
P0
P
a
xy
z
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Lines and Vectors
We can describe lines in R3 by referring to vectors in V3.
Consider a nonzero vector a and a point P0 = (x0, y0, z0). Thevector with initial point P0 in the direction of a = a1, a2, a3 is
P0P = tawhere t is a scalar.
If point P = (x, y, z) then
x x0, y y0, z z0 = ta1, a2, a3.
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Parametric and Symmetric Equations
The parametric equations for the line through (x0, y0, z0) inthe direction of vector a = a1, a2, a3:
x = x0 + ta1
y = y0 + ta2
z = z0 + ta3
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Parametric and Symmetric Equations
The parametric equations for the line through (x0, y0, z0) inthe direction of vector a = a1, a2, a3:
x = x0 + ta1
y = y0 + ta2
z = z0 + ta3
The symmetric equations for the line are
x
x0
a1 =y
y0
a2 =z
z0
a3 .
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Example (1 of 3)
Example
Let P0 = (5, 1, 3) and a = i + 4j 2k and find the equation ofthe line through P0 in the direction of a in parametric andsymmetric form.
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Example (1 of 3)
Example
Let P0 = (5, 1, 3) and a = i + 4j 2k and find the equation ofthe line through P0 in the direction of a in parametric andsymmetric form.
Parametric Form:
x = 5 + t
y = 1 + 4t
z = 3 2t
Symmetric Form:
x 11
=y 1
4=
z 32
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E l (2 f 3)
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Example (2 of 3)
Example
Find the equation of the line which passes through the points
A = (2, 4,3) and B = (3,1, 1). Where does this lineintersect the xy-plane?
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E l (2 f 3)
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Example (2 of 3)
Example
Find the equation of the line which passes through the points
A = (2, 4,3) and B = (3,1, 1). Where does this lineintersect the xy-plane?
The line is parallel to the vector:
v = AB = 3 2,1 4, 1 (3) = 1,5, 4.
J. Robert Buchanan Lines and Planes in Space
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Example (2 of 3)
Example
Find the equation of the line which passes through the points
A = (2, 4,3) and B = (3,1, 1). Where does this lineintersect the xy-plane?
The line is parallel to the vector:
v = AB = 3 2,1 4, 1 (3) = 1,5, 4.The parametric form of the line is:
x = 2 ty = 4
5t
z = 3 + 4t
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Example (2 of 3)
Example
Find the equation of the line which passes through the points
A = (2, 4,3) and B = (3,1, 1). Where does this lineintersect the xy-plane?
The line is parallel to the vector:
v = AB = 3 2,1 4, 1 (3) = 1,5, 4.The parametric form of the line is:
x = 2 ty = 4
5t
z = 3 + 4tThe line intersects the xy-plane when z = 0 which implies
t = 3/4.
The point of intersection is (x, y, z) = (5/4, 1/4, 0).J. Robert Buchanan Lines and Planes in Space
Example (3 of 3)
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Example (3 of 3)
Example
Do the following two lines intersect?
x = 1 + t, y = 2 + 3t, z = 4 t
x = 2s, y = 3 + s, z = 3 + 4s
J. Robert Buchanan Lines and Planes in Space
Example (3 of 3)
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Example (3 of 3)
Example
Do the following two lines intersect?
x = 1 + t, y = 2 + 3t, z = 4 t
x = 2s, y = 3 + s, z = 3 + 4s
If the lines intersect the lines must have a point in common.Solving the system of two equations in two unknowns:
1 + t = 2s
2 + 3t = 3 + s
implies s = 8/5 and t = 11/5.
J. Robert Buchanan Lines and Planes in Space
Example (3 of 3)
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Example (3 of 3)
Example
Do the following two lines intersect?
x = 1 + t, y = 2 + 3t, z = 4 t
x = 2s, y = 3 + s, z = 3 + 4s
If the lines intersect the lines must have a point in common.Solving the system of two equations in two unknowns:
1 + t = 2s
2 + 3t = 3 + s
implies s = 8/5 and t = 11/5.
However, using these s and t values makes the z-coordinates
unequal. Thus the lines do not intersect.
J. Robert Buchanan Lines and Planes in Space
Parallel and Orthogonal Lines
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Parallel and Orthogonal Lines
Definition
Let l1 and l2 be two lines in R3, with parallel vectors a and b,
respectively, and let be the angle between a and b.
1 The lines l1 and l2 are parallel whenever a and b are
parallel.2 If l1 and l2 intersect, then
the angle between l1 and l2 is andthe lines l1 and l2 are orthogonal whenever a and b areorthogonal.
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Planes in R3
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Planes in R
A plane can be thought of as the collection of all lines
orthogonal to a given line.
x
y
z
J. Robert Buchanan Lines and Planes in Space
Planes and Vectors
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Planes and Vectors
If point P0 = (x0, y0, z0) lies in the plane and vectora = a1, a2, a3 is normal to the plane (i.e., orthogonal to everyline in the plane) and if point P = (x, y, z) is an arbitrary point inthe plane, then
P0Pis orthogonal to
a,
J. Robert Buchanan Lines and Planes in Space
Planes and Vectors
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Planes and Vectors
If point P0 = (x0, y0, z0) lies in the plane and vectora = a1, a2, a3 is normal to the plane (i.e., orthogonal to everyline in the plane) and if point P = (x, y, z) is an arbitrary point inthe plane, then
P
0P is orthogonal to a,
x x0, y y0, z z0 a1, a2, a3 = 0, and
J. Robert Buchanan Lines and Planes in Space
Planes and Vectors
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Planes and Vectors
If point P0 = (x0, y0, z0) lies in the plane and vectora = a1, a2, a3 is normal to the plane (i.e., orthogonal to everyline in the plane) and if point P = (x, y, z) is an arbitrary point inthe plane, then
P
0P is orthogonal to a,
x x0, y y0, z z0 a1, a2, a3 = 0, andthe equation of the plane is
a1(x
x0) + a2(y
y0) + a3(z
z0) = 0.
J. Robert Buchanan Lines and Planes in Space
Example (1 of 2)
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Example (1 of 2)
Example
Find the equation of the plane through the point (2, 4,1) withnormal vector n = 2, 3, 4.
J. Robert Buchanan Lines and Planes in Space
Example (1 of 2)
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Example (1 of 2)
Example
Find the equation of the plane through the point (2, 4,1) withnormal vector n = 2, 3, 4.
x 2, y 4, z (1) 2, 3, 4 = 02x + 3y + 4z = 12
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Example (2 of 2)
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p ( )
Example
Find the equation of the plane containing the points
P = (1, 3, 2), Q = (3,1, 6), and R = (5, 2, 0).
J. Robert Buchanan Lines and Planes in Space
Example (2 of 2)
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p ( )
Example
Find the equation of the plane containing the points
P = (1, 3, 2), Q = (3,1, 6), and R = (5, 2, 0).First we must find a vector orthogonal to the plane containing
the three points.
J. Robert Buchanan Lines and Planes in Space
Example (2 of 2)
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p ( )
Example
Find the equation of the plane containing the points
P = (1, 3, 2), Q = (3,1, 6), and R = (5, 2, 0).First we must find a vector orthogonal to the plane containing
the three points.
Let a = PQ = 2,4,4 and let b = PR = 4,1,2, thenusing the cross product we have a vector perpendicular to the
plane.
n = a b = 12, 20, 14
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Example (2 of 2)
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p ( )
Example
Find the equation of the plane containing the points
P = (1, 3, 2), Q = (3,1, 6), and R = (5, 2, 0).First we must find a vector orthogonal to the plane containing
the three points.
Let a = PQ = 2,4,4 and let b = PR = 4,1,2, thenusing the cross product we have a vector perpendicular to the
plane.
n = a b = 12, 20, 14
The equation of the plane is
x 1, y 3, z 2 12, 20, 14 = 06x + 10y + 7z = 50
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Remarks
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The equation of a plane in R3 has the form:
ax + by + cz = d
where not all of a, b, and c can be zero.
J. Robert Buchanan Lines and Planes in Space
Remarks
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The equation of a plane in R3 has the form:
ax + by + cz = d
where not all of a, b, and c can be zero.
If ax + by + cz = d defines a plane, then v = a, b, c isnormal to the plane.
J. Robert Buchanan Lines and Planes in Space
Remarks
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The equation of a plane in R3 has the form:
ax + by + cz = d
where not all of a, b, and c can be zero.
If ax + by + cz = d defines a plane, then v = a, b, c isnormal to the plane.
An easy method for sketching a plane is to sketch the
simplex of the plane defined by its intersections with the
coordinate axes.
J. Robert Buchanan Lines and Planes in Space
Parallel and Orthogonal Planes
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Definition
Two planes with normal vectors a and b are
1 parallel if a and b are parallel.2 orthogonal if a and b are orthogonal.
J. Robert Buchanan Lines and Planes in Space
Example (1 of 3)
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Example
Are the planes defined by x + 2y 3z = 4 and2x + 4y 6z = 1 parallel?
J. Robert Buchanan Lines and Planes in Space
Example (1 of 3)
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Example
Are the planes defined by x + 2y 3z = 4 and2x + 4y 6z = 1 parallel?
A normal vector to the first plane is a = 1, 2,3 while anormal vector to the second plane is b = 2, 4,6.
J. Robert Buchanan Lines and Planes in Space
Example (1 of 3)
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Example
Are the planes defined by x + 2y 3z = 4 and2x + 4y 6z = 1 parallel?
A normal vector to the first plane is a = 1, 2,3 while anormal vector to the second plane is b = 2, 4,6.Since b is a scalar multiple of a (namely b = 2a) then thenormal vectors are parallel, which implies the original planes
are parallel.
J. Robert Buchanan Lines and Planes in Space
Example (2 of 3)
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Example
Find the angle between the planes
x + y + z = 1
x 2y + 3z = 2.
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Example (2 of 3)
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Example
Find the angle between the planes
x + y + z = 1
x 2y + 3z = 2.The angle between the planes will be the angle between their
normal vectors.
J. Robert Buchanan Lines and Planes in Space
Example (2 of 3)
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Example
Find the angle between the planes
x + y + z = 1
x 2y + 3z = 2.The angle between the planes will be the angle between their
normal vectors.
Let a = 1, 1, 1 and b = 1,2, 3, then
a
b =
a
b
cos
2 = 42 cos 1.25707 72.02
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Example (3 of 3)
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Example
Find the line of intersection of the two planes
x + y + z = 1
x 2y + 3z = 2.
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Example (3 of 3)
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Example
Find the line of intersection of the two planes
x + y + z = 1
x 2y + 3z = 2.Eliminate x from the two equations and then treat z as the
parameter.1 y z = x = 2 + 2y 3z
y = 13
+2
3z
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Example (3 of 3)
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Example
Find the line of intersection of the two planes
x + y + z = 1
x 2y + 3z = 2.Eliminate x from the two equations and then treat z as the
parameter.1 y z = x = 2 + 2y 3z
y = 13
+2
3z
Parametric Form:x =
4
3 5
3t
y = 13
+2
3t
z = tJ. Robert Buchanan Lines and Planes in Space
Graph
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2
1
0
1
2
x
2
1
0
1
2
y
2
1
0
1
2
z
J. Robert Buchanan Lines and Planes in Space
Distance from a Point to a Plane
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P1
P2
a
The distance from P2 to the plane is compaP1P2.
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Distance from a Point to a Plane
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If a=
a,
b,
c
, P1 = (
x1,
y1,
z1)
, and P2 = (
x2,
y2,
z2)
then
compa
P1P2 =
a P1P2a
= a, b, c
x2
x1, y2
y1, z2
z1
a2 + b2 + c2=
ax2 + by2 + cz2 (ax1 + by1 + cz1)a2 + b2 + c2
distance =|ax2 + by2 + cz2 + d|
a2 + b2 + c2
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Example (1 of 3)
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Example
Find the distance from ( 12 , 0, 1) to 5x + y z = 1.
Rather than trying to apply the distance formula from memory it
may be easier to recall that the distance from the point to the
plane is |compaP1P2|, where a is the normal vector to the planeand we are free to pick point P1 to be any point in the plane.
J. Robert Buchanan Lines and Planes in Space
Example (1 of 3)
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Example
Find the distance from ( 12 , 0, 1) to 5x + y z = 1.
Rather than trying to apply the distance formula from memory it
may be easier to recall that the distance from the point to the
plane is |compaP1P2|, where a is the normal vector to the planeand we are free to pick point P1 to be any point in the plane.
Let P2 = (12 , 0, 1), let P1 = (0, 1, 0), and a = 5, 1,1, then
|compaP1P2| = 1/2,
1, 1
5, 1,
1
5, 1,1 =1
63 .
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Example (2 of 3)
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Example
Find the distance between the planes
10x + 2y 2z = 55x + y z = 1
If the planes were not parallel we could immediately declare the
distance between them is 0.
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Example (2 of 3)
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Example
Find the distance between the planes
10x + 2y 2z = 55x + y z = 1
If the planes were not parallel we could immediately declare the
distance between them is 0.
Pick a point in the second plane, say P2 = (0, 1, 0) and find itsdistance to the first plane.
J. Robert Buchanan Lines and Planes in Space
Example (2 of 3)
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Example
Find the distance between the planes
10x + 2y 2z = 55x + y z = 1
If the planes were not parallel we could immediately declare the
distance between them is 0.
Pick a point in the second plane, say P2 = (0, 1, 0) and find itsdistance to the first plane.
Note that P1 = (1/2, 0, 0) is in the first plane and the normal
vector to the first plane is a = 10, 2,2.
|compaP1P2| =
1/2, 1, 0 10, 2,210, 2,2
=1
2
3
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Example (3 of 3)
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Example
Find the distance between the skew lines
x = 1 + t, y = 2 + 3t, z = 4 t, and
x = 2s, y = 3 + s, z = 3 + 4s.
J. Robert Buchanan Lines and Planes in Space
Example (3 of 3)
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Example
Find the distance between the skew lines
x = 1 + t, y = 2 + 3t, z = 4 t, and
x = 2s, y = 3 + s, z = 3 + 4s.We must find the equations of two parallel planes, each
containing one of the lines above.
J. Robert Buchanan Lines and Planes in Space
Example (3 of 3)
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Example
Find the distance between the skew lines
x = 1 + t, y = 2 + 3t, z = 4 t, and
x = 2s, y = 3 + s, z = 3 + 4s.We must find the equations of two parallel planes, each
containing one of the lines above.
The common normal vector shared by the planes must be
perpendicular to both lines.
a =
1, 3,
1
2, 1, 4
=13,
6,
5
J. Robert Buchanan Lines and Planes in Space
Example (3 of 3)
Example
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Example
Find the distance between the skew lines
x = 1 + t, y = 2 + 3t, z = 4 t, and
x = 2s, y = 3 + s, z = 3 + 4s.We must find the equations of two parallel planes, each
containing one of the lines above.
The common normal vector shared by the planes must be
perpendicular to both lines.
a =
1, 3,
1
2, 1, 4
=13,
6,
5
Let P1 = (1,2, 4) be a point on the first line and letP2 = (0, 3,3) be a point on the second line.
|compa
P1P2
|=
1, 5,7 13,6,5
13,6,5 =
32
115J. Robert Buchanan Lines and Planes in Space
Graph
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3.0
3.5
4.0
4.5
5.0
y
1.0 1.5 2.0 2.5 3.0
x
1.0
1.5
2.0
2.5
3.0
z
J. Robert Buchanan Lines and Planes in Space
Homework
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Read Section 10.5.
Exercises: 167 odd.
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