Download
# Chapter Section Lines and Planes in Space Example Show that the line through the points and is perpendicular to the line through the points and PDF document - DocSlides

conchita-marotz | 2014-12-12 | General

### Presentations text content in Chapter Section Lines and Planes in Space Example Show that the line through the points and is perpendicular to the line through the points and

Show

Page 1

Chapter 12 Section 5 Lines and Planes in Space

Page 2

Example 1 Show that the line through the points (0 1) and (1 6) is perpendicular to the line through the points ( 1) and ( 2) . Vector equation for the ﬁrst line: t < Vector equation for the second line: s < cos || (1)(3) + ( 2)(4) + (5)(1) + ( 2) + 5 + 4 + 1 30 26 Remark: These two lines are skew

Page 3

Example 2 (a) Find parametric equations for the line through (5 0) that is perpendicular to the plane = 1 A normal vector to the plane is: ) = t < (b) In what points does this line intersect the coordinate planes? xy -plane: 0 = 0 + = 0 (0) = yz -plane: 0 = 5 + ) = zx -plane: 0 = 1 + 1) = 1 (1) =

Page 4

Example 3 Parallelism, intersection for: ) = t < 1 + 2 t,t, 1 + 4 t > + 2 + 2 ) = s < < s, 2 + 2 s, 2 + 3 s > 1 + 2 t,t, 1 + 4 t > < s, 2 + 2 s, 2 + 3 s >

Page 5

1 + 2 2 + 2 1 + 4 2 + 3 Solving the ﬁrst two equations: = 0 , s = 1 Checking the third equation: 1 + 4(0) = 2 + 3(1) (satisﬁed) Consequently: (0) (1)

Page 6

Example 4 Plane through (2 3) (5 4) (2 4) (2 3) (5 4) = (2 3) (2 4) = 21 Equation for plane: = 0 2)(7) + ( 1)( 21) + ( + 3)( 9) = 0 21 + (( 2)(7) + ( 1)( 21) + (3)( 9)) = 0 21 = 20

Page 7

Example 5 Plane through the point ( 1) and the line = 5 t, y = 1 + t, z ) = t < (0) = 1) (0 0) = Equation for plane: = 0 1))(0) + ( 0)( 4) + ( 1)( 4) = 0 (( 1)(0) + (0)( 4) + (1)( 4)) = 0 + 4 = 0 = 1

Page 8

Example 6 Intersection of line and plane: Line: = 1 t, y t, z = 1 + Plane: = 1 Substitute line in plane equation: (1 + ) = 1 2(1 ) + ( 0 = + 1 2 + 2 2 = 2 = 1 Line Plane = 1 + 1

Page 9

Example 7 Direction numbers for intersection of planes: Plane 1: = 1 Plane 2: = 0 Line direction numbers: Unit vector:

Page 10

Example 8 Intersection of planes: Plane 1: = 1 Plane 2: 2 = 1 Line direction numbers: Common point: (0 1) Symmetric equations:

Page 11

Example 9 Plane of points equidistant from (1 0) (0 1) Midpoint = Normal = Equation: = 0 )(1) + ( 1)(0) + ( )( 1) = 0 ) = 0 = 0 The plane has the equation

Page 12

Example 10 Find an equation for the plane with -intercept a, y -intercept b, z -intercept Given points in plane: = ( a, 0) = (0 ,b, 0) = (0 ,c < a, c > ,b, c > Normal < bc,ca,ab > Equation: 0 = ( ,c > 0 = < x,y,z c > < bc,ca,ab > 0 = bcx cay ab abc bcx cay abz

Page 13

If = 0 : = ( abc 1 = bc abc ca abc ab abc 1 = ax by cz

Page 14

Example 11 Find parametric equations for the line through (0 2) that is parallel to the plane = 2 and perpendicular to the line = 1 + t, y = 1 t, z = 2 t. Principal task = Find direction of Let < a,b,c > (Plane) (Normal(Plane)) Direction(Line)) Let Vector equation of ) = t < Parametric equations of = 3 t, y = 1 t, z = 2

Page 15

Example 12 Find equations of the planes parallel to the plane + 2 = 1 and two units away from it. The distance between parallel planes ax by cz = 0 and ax by cz = 0 is < a,b,c > + 2 + ( 2) = 3 1) = 2 1) = 6 = ( 1) 6 = + 2 + 5 = 0 + 2 7 = 0

Page 16

Example 13 Line Direction Line + 1 = Direction x,x,x ) has + 1 = No solution! and are skew. Cross product: and and Then: Distance , L ) = Distance , P

Page 17

(1 1) < x,y,z > = 0 + 0 = 0 (0 3) < x,y,z > = 0 + 1 = 0 Distance formula: (1) + ( 2) + (1)

Page 18

Example 14 Geometric descriptions (a) c, c real: Family of planes orthogonal to the line z. (b) cz = 1 , c real: Family of planes containing the line = 1 ,z = 0 Plane is vertical, if = 0 Else, plane has -intercept (c) cos sin = 1 , real: Family of planes parallel to -axis, orthogonal to cos θ, sin θ >, containing the point (0 cos θ, sin Alternatively: Family of planes parallel to -axis, tangent to the cylinder = 1 For given the plane contains the point (0 cos θ, sin

Vector equation for the 64257rst line t Vector equation for the second line s cos 13 24 51 2 5 4 1 30 26 Remark These two lines are skew brPage 3br Example 2 a Find parametric equations for the line through 5 0 that is perpendicular to th ID: 23046

- Views :
**176**

**Direct Link:**- Link:https://www.docslides.com/conchita-marotz/chapter-section-lines-and-planes
**Embed code:**

Download this pdf

DownloadNote - The PPT/PDF document "Chapter Section Lines and Planes in Sp..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

Page 1

Chapter 12 Section 5 Lines and Planes in Space

Page 2

Example 1 Show that the line through the points (0 1) and (1 6) is perpendicular to the line through the points ( 1) and ( 2) . Vector equation for the ﬁrst line: t < Vector equation for the second line: s < cos || (1)(3) + ( 2)(4) + (5)(1) + ( 2) + 5 + 4 + 1 30 26 Remark: These two lines are skew

Page 3

Example 2 (a) Find parametric equations for the line through (5 0) that is perpendicular to the plane = 1 A normal vector to the plane is: ) = t < (b) In what points does this line intersect the coordinate planes? xy -plane: 0 = 0 + = 0 (0) = yz -plane: 0 = 5 + ) = zx -plane: 0 = 1 + 1) = 1 (1) =

Page 4

Example 3 Parallelism, intersection for: ) = t < 1 + 2 t,t, 1 + 4 t > + 2 + 2 ) = s < < s, 2 + 2 s, 2 + 3 s > 1 + 2 t,t, 1 + 4 t > < s, 2 + 2 s, 2 + 3 s >

Page 5

1 + 2 2 + 2 1 + 4 2 + 3 Solving the ﬁrst two equations: = 0 , s = 1 Checking the third equation: 1 + 4(0) = 2 + 3(1) (satisﬁed) Consequently: (0) (1)

Page 6

Example 4 Plane through (2 3) (5 4) (2 4) (2 3) (5 4) = (2 3) (2 4) = 21 Equation for plane: = 0 2)(7) + ( 1)( 21) + ( + 3)( 9) = 0 21 + (( 2)(7) + ( 1)( 21) + (3)( 9)) = 0 21 = 20

Page 7

Example 5 Plane through the point ( 1) and the line = 5 t, y = 1 + t, z ) = t < (0) = 1) (0 0) = Equation for plane: = 0 1))(0) + ( 0)( 4) + ( 1)( 4) = 0 (( 1)(0) + (0)( 4) + (1)( 4)) = 0 + 4 = 0 = 1

Page 8

Example 6 Intersection of line and plane: Line: = 1 t, y t, z = 1 + Plane: = 1 Substitute line in plane equation: (1 + ) = 1 2(1 ) + ( 0 = + 1 2 + 2 2 = 2 = 1 Line Plane = 1 + 1

Page 9

Example 7 Direction numbers for intersection of planes: Plane 1: = 1 Plane 2: = 0 Line direction numbers: Unit vector:

Page 10

Example 8 Intersection of planes: Plane 1: = 1 Plane 2: 2 = 1 Line direction numbers: Common point: (0 1) Symmetric equations:

Page 11

Example 9 Plane of points equidistant from (1 0) (0 1) Midpoint = Normal = Equation: = 0 )(1) + ( 1)(0) + ( )( 1) = 0 ) = 0 = 0 The plane has the equation

Page 12

Example 10 Find an equation for the plane with -intercept a, y -intercept b, z -intercept Given points in plane: = ( a, 0) = (0 ,b, 0) = (0 ,c < a, c > ,b, c > Normal < bc,ca,ab > Equation: 0 = ( ,c > 0 = < x,y,z c > < bc,ca,ab > 0 = bcx cay ab abc bcx cay abz

Page 13

If = 0 : = ( abc 1 = bc abc ca abc ab abc 1 = ax by cz

Page 14

Example 11 Find parametric equations for the line through (0 2) that is parallel to the plane = 2 and perpendicular to the line = 1 + t, y = 1 t, z = 2 t. Principal task = Find direction of Let < a,b,c > (Plane) (Normal(Plane)) Direction(Line)) Let Vector equation of ) = t < Parametric equations of = 3 t, y = 1 t, z = 2

Page 15

Example 12 Find equations of the planes parallel to the plane + 2 = 1 and two units away from it. The distance between parallel planes ax by cz = 0 and ax by cz = 0 is < a,b,c > + 2 + ( 2) = 3 1) = 2 1) = 6 = ( 1) 6 = + 2 + 5 = 0 + 2 7 = 0

Page 16

Example 13 Line Direction Line + 1 = Direction x,x,x ) has + 1 = No solution! and are skew. Cross product: and and Then: Distance , L ) = Distance , P

Page 17

(1 1) < x,y,z > = 0 + 0 = 0 (0 3) < x,y,z > = 0 + 1 = 0 Distance formula: (1) + ( 2) + (1)

Page 18

Example 14 Geometric descriptions (a) c, c real: Family of planes orthogonal to the line z. (b) cz = 1 , c real: Family of planes containing the line = 1 ,z = 0 Plane is vertical, if = 0 Else, plane has -intercept (c) cos sin = 1 , real: Family of planes parallel to -axis, orthogonal to cos θ, sin θ >, containing the point (0 cos θ, sin Alternatively: Family of planes parallel to -axis, tangent to the cylinder = 1 For given the plane contains the point (0 cos θ, sin

Today's Top Docs

Related Slides