Calculus 3 Vector

8/6/2019 Calculus 3 Vector

1/55

Lines and Planes in SpaceMATH 311, Calculus III

J. Robert Buchanan

Department of Mathematics

Summer 2010

J. Robert Buchanan Lines and Planes in Space
http://find/http://goback/


2/55

Lines and Points

P0

P

a

xy

z



3/55

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.



4/55

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



5/55

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 .



6/55

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.



7/55

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


E l (2 f 3)


8/55

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?


E l (2 f 3)


9/55

Example (2 of 3)

Example



The line is parallel to the vector:

v = AB = 3 2,1 4, 1 (3) = 1,5, 4.


E l (2 f 3)


10/55

Example (2 of 3)

Example




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


E l (2 f 3)


11/55

Example (2 of 3)

Example




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)


12/55

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


Example (3 of 3)


13/55

Example (3 of 3)

Example


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.


Example (3 of 3)


14/55

Example (3 of 3)

Example


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.


Parallel and Orthogonal Lines


15/55

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.



16/55

Planes in R3


17/55

Planes in R

A plane can be thought of as the collection of all lines

orthogonal to a given line.

x

y

z


Planes and Vectors


18/55

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,


Planes and Vectors


19/55

Planes and Vectors


P

0P is orthogonal to a,

x x0, y y0, z z0 a1, a2, a3 = 0, and


Planes and Vectors


20/55

Planes and Vectors


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.


Example (1 of 2)


21/55

Example (1 of 2)

Example

Find the equation of the plane through the point (2, 4,1) withnormal vector n = 2, 3, 4.


Example (1 of 2)


22/55

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


Example (2 of 2)


23/55

p ( )

Example

Find the equation of the plane containing the points

P = (1, 3, 2), Q = (3,1, 6), and R = (5, 2, 0).


Example (2 of 2)


24/55

p ( )

Example


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.


Example (2 of 2)


25/55

p ( )

Example



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


Example (2 of 2)


26/55

p ( )

Example



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


Remarks


27/55

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.


Remarks


28/55


ax + by + cz = d


If ax + by + cz = d defines a plane, then v = a, b, c isnormal to the plane.


Remarks


29/55


ax + by + cz = d


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.


Parallel and Orthogonal Planes


30/55

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.


Example (1 of 3)


31/55

Example

Are the planes defined by x + 2y 3z = 4 and2x + 4y 6z = 1 parallel?


Example (1 of 3)


32/55

Example


A normal vector to the first plane is a = 1, 2,3 while anormal vector to the second plane is b = 2, 4,6.


Example (1 of 3)


33/55

Example


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.


Example (2 of 3)


34/55

Example

Find the angle between the planes

x + y + z = 1

x 2y + 3z = 2.


Example (2 of 3)


35/55

Example


x + y + z = 1

x 2y + 3z = 2.The angle between the planes will be the angle between their

normal vectors.


Example (2 of 3)


36/55

Example


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


Example (3 of 3)


37/55

Example

Find the line of intersection of the two planes

x + y + z = 1

x 2y + 3z = 2.


Example (3 of 3)


38/55

Example


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


Example (3 of 3)


39/55

Example


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


40/55

2

1

0

1

2

x

2

1

0

1

2

y

2

1

0

1

2

z


Distance from a Point to a Plane


41/55

P1

P2

a

The distance from P2 to the plane is compaP1P2.


Distance from a Point to a Plane


42/55

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



43/55

Example (1 of 3)


44/55

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.


Example (1 of 3)


45/55

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 .



46/55

Example (2 of 3)


47/55

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.


Example (2 of 3)


48/55

Example


10x + 2y 2z = 55x + y z = 1



Pick a point in the second plane, say P2 = (0, 1, 0) and find itsdistance to the first plane.


Example (2 of 3)


49/55

Example


10x + 2y 2z = 55x + y z = 1



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


Example (3 of 3)

E l


50/55

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.


Example (3 of 3)

E l


51/55

Example


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.


Example (3 of 3)

E l


52/55

Example


x = 1 + t, y = 2 + 3t, z = 4 t, and



The common normal vector shared by the planes must be

perpendicular to both lines.

a =

1, 3,

1

2, 1, 4

=13,

6,

5


Example (3 of 3)

Example


53/55

Example


x = 1 + t, y = 2 + 3t, z = 4 t, and



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


54/55

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


Homework


55/55

Read Section 10.5.

Exercises: 167 odd.


Date post:	07-Apr-2018
Category:	Documents
Upload:	thavalong-prum
View:	278 times
Download:	3 times

Calculus 3 Vector

Documents