Math 126: Calculus III Section 12.5: Equation of Lines and Planes

Transcription

1 1 Math 16: Calculus III Section 1.5: Equation of Lines and Planes

2 Vector equations of lines Consider the position vector r 0 (which gives points to a point on the line) and the direction v line). Additional points on the line can be reached by multiplying the vector v by the scaling v tv r 0 Origin r 0 Origin

3 3 Vector equations of lines, continued Thus, a line in n through a point given by the position vector r0 and in the direction of v ca r r 0 tv (where t is a parameter). line r 0 r r 0 tv Origin

4 4 Vector equations of lines, continued So, when we allow tto vary, we are left with a line as below (animation)

5 5 Lines: Parametric Equations So, if r 0 is a position vector (for a point on the line), v is a direction vector for the line, and r (where t is a parameter), then: r 0 x 0, y 0, z 0 v a, b, c and r x, y, z then x, y, z x 0 t a, y 0 t b, z 0 t c is the vector equation of a line L parallel to the vector v. Or, the parametric equations are: x x 0 t a; y y 0 t b; z

6 6 Example 1 Find the parametric equations of the line through 1,, 3 and parallel to 4, 5, 6.

7 7 Eliminating the Parameter Eliminating the parameter leads to what are known as symmetric equations. That is, assumin equation in (1.) x x 0 t a; y y 0 t b; z z 0 t c for the parameter t. For example: t xx 0 a Setting the three equations equal gives the symmetric equation (.) xx 0 a yy 0 b zz 0 c when a, b, c 0 In the case where v has a zero component in a given direction (e.g., a 0), then the symmetr yy 0 zz 0 when b, c 0. This is a line on the plane x x 0. (Note: these symmetric b c (1.) rather than from the symmetric equations (.)).

8 8 Example a.) Find the symmetric equations of the line from example 1. b.) Find the point were the line intersects the xy plane (where z 0)

9 9 The Equation of Line between Points: The Picture We are trying to find the line between two points. In the illustration, the points would be r 0 green line represents the line segment between the two points (on 0 t 1) and the blue line for all values of t. r 1 7,1, r 0 6,6,5

10 1 The Equation of Line between Points: The Formula To describe the line from r 0 to r 1, begin with 1r 0 0r 1 and go to 0r 0 1r 1. T equation: r 1tr 0 t r 1, 0 t 1. This is a powerful formula - understand the derivation rather than memorizing it.

11 1 Example 3 Find the parametric representation of a line from 1,, 3 to3, 7, 11.

12 1 Planes in 3. A plane can be determined by a point and a vector orthogonal to the plane (a normal vector). Why? Notice that this is similar to finding a line by having its slope and a point.

13 1 Forming a Plane - Seeing Dimension The following animation shows two position vectors r 0 and r of points on a plane, the vecto a normal vector to the plane n. The purpose of the animation is to help the student see the dim r Origin r r 0 r 0 n

14 1 Forming a Plane - the Normal Vector The following graph shows two position vectors r 0 and r of points on a plane, the vector r normal vector to the plane n. The origin and z-axis are shown to help the student visualize di orthogonal to the vector r r 0. n r 0 Origin r r 0 r Thus, n r r 0 0.

15 1 The Equation of a Plane Let n, r, and r 0 be defined as follows: n a, b, c (normal vector) r x, y, z (position vector for an arbitrary point on the plane) r 0 x 0, y 0, z 0 (position vector for a given point on the plane)s Since n r r 0 0, we have 0 a, b, c x, y, z x 0, y 0, z 0 0 a, b, c xx 0, yy 0, zz 0 0 axx 0 byy 0 czz 0 This can also be written as axbyczd 0

16 1 Example 4 Find the equation of the plane that includes the point 1,, 3 and that has normal vector

17 1 Question How would we find the equation of the plane through three points (assuming that they are no

18 1 Definition We define the angle between planes to be the angle between their normal vectors.

19 1 Example 5 Find the parametric equations for the line Lof intersection of the planes z xyand x Use the following steps. 1.) Find normal vectors to each plane..) Find the cross product of the normals to determine the direction of the line L.

20 3.) Determine a point on the line L (one method would be to set z 0). In this case, when z0, we are left with x ) Express your answer. and y 1 3. Or, the line goes thru 1, 1 3 3,

21 Note There are an infinite number of vectors normal to a plane at a point.

22 Distance from a Plane to a Point - The Scenario To find the distance D from a plane to a fixed point, we need to to find determine the magnitu plane and whose tip is at the point. As we have seen, finding normal vectors to a plane is not difficult. Unfortunately, the norma axbyczd 0 only gives the direction from the plane to the point, not the magnitude P 1 x 1,y 1,z 1 n

23 Distance from a Plane to a Point - The Picture Deriv To find the distance D from the plane axbyczd 0 to the point P 1, we will pick so ax 0 by 0 cz 0 d 0) and find the magnitude of the projection of the vector v from P 0 P 1 x 1,y 1,z 1 v n P 0 x 0,y 0,z 0

24 Distance from a Plane to a Point - The Derivation We have the following vectors: n a, b, c (the normal to the plane axbyczd 0) v x 1 x 0, y 1 y 0, z 1 z 0 (a vector from P 0 to P 1 ) To find the distance D from the plane to P 1, we need to find: D comp n v D n v n D ax 1 x 0 by 1 y 0 cz 1 z 0 n D ax 1 ax 0 by 1 by 0 cz 1 cz 0 n D ax 1 ax 0 by 1 by 0 cz 1 cz 0 dd (add 0 dd) n D ax 1 ax 0 by 1 by 0 cz 1 cz 0 dd n (recall that ax 0 by 0 cz 0 d 0 since P 0 is on D ax 1 by 1 cz 1 d n

25 Mathematica Scratch Work: Lines in 3 Needs"VisualLA`" $TextStyle FontSize 1; $FormatType TraditionalForm; WINDOW : 0, 10, 0, 10, 0, 10; Origin 0, 0, 0; A 1,, 3; B 6, 5, 4; V BA; ShowDrawVector3DA, DisplayFunction Identity, DrawVector3DV, DisplayFunction Identity, PlotRange W DisplayFunction $DisplayFunction; TableShowDrawVector3DA t V, DisplayFunction Identity, DisplayFunction $DisplayFunction, PlotRange WINDOW, Ticks t, 0, 1.5, 0.05; TableShowParametricPlot3DA s B, s, 0.05, t, DisplayFunct DrawVector3DA t B, DisplayFunction Identity, DisplayF PlotRange WINDOW, Ticks False, t, 0, 1.5, 0.05; TableShowGraphics3DLineA, A t V, DrawVector3DA, Dis DrawVector3DA t V, DisplayFunction Identity, DisplayF PlotRange WINDOW, Ticks False, t, 0.05, 1.5, 0.05; WINDOW :, 10, 0, 10, 0, 5;

26 t 1.5; P1 : Show P : Graphics3DText"Origin", 0, 0, 0, 0, 1, Text"r 0", A Text"v ", A V, 0, 1, DrawVector3DA, DisplayFun DrawVector3DA, A V, DisplayFunction Identity, Plo Ticks False, Boxed False; Show Graphics3DText"Origin", 0, 0, 0, 0, 1, Text"r 0", A Text"tv ", A 1.5 V, 0, 1, DrawVector3DA, Displ DrawVector3DA, A t V, DisplayFunction Identity, P Ticks False, Boxed False; ShowGraphicsArrayP1, P, DisplayFunction $DisplayFunctio P3 : P3; Show Graphics3DText"Origin", 0, 0, 0, 0, 1, Text"r 0", A Text"line", A 1.5 V, 0, 1, Text"r r 0tv ", A t V, DrawVector3DA, DisplayFunction Identity, DrawVector3DA t V, DisplayFunction Identity, PlotRa Ticks False, Boxed False;

27 WINDOW : 3, 3, 4, 3, 40, 0; fx_, y_ : xy0; P4 : Plot3Dfx, y, x, 3, 3, y, 3, 3, Mesh False, Displa A,, f, ; B, 1, f, 1; NORMAL 1,, 1; t 1; Table Show P4, Graphics3DLineA, B, Line0, 0, 40, 0, 0, 0 Text"Origin", 0, 0, 0, 1, 1, Text"r 0", A, 4, 0, Text"r r 0", A B, 0, 1, Text"n ", A NORMAL,, 0 DrawVector3DA, HeadLength 0.1, HeadAngle 0.05, Displa DrawVector3DB, HeadLength 0.1, HeadAngle 0.05, Displa DrawVector3DA, A NORMAL, HeadLength 0.0, HeadAngle DisplayFunction Identity, PlotRange WINDOW, Ticks F ViewPoint 3, s, 5, DisplayFunction $DisplayFunction, WINDOW : 0, 4, 8, 1, 40, 0; fx_, y_ : xy0; P4 : Plot3Dfx, y, x, 0, 4, y, 8, 1, Mesh False, DisplayF A,, f, ; B 3, 5, f3, 5; NORMAL 1,, 1; t 1; ShowP4, PointPlot3DA, A NORMAL, DisplayFunction Identi Graphics3DText"P 1 x 1,y 1,z 1 ", A NORMAL, 1.1, 0, Text DrawVector3DA, A NORMAL, HeadLength 0.0, HeadAngle DisplayFunction Identity, PlotRange WINDOW, Ticks Fa ViewPoint 3, 3, 5, DisplayFunction $DisplayFunction;

28 WINDOW : 0, 4, 8, 1, 40, 0; fx_, y_ : xy0; P4 : Plot3Dfx, y, x, 0, 4, y, 8, 1, Mesh False, DisplayF A,, f, ; B 3, 5, f3, 5; NORMAL 1,, 1; t 1; ShowP4, PointPlot3DA, A NORMAL, B, DisplayFunction Ide Graphics3DDashing0.01, 0.01, LineA, B, LineA N Text"P 0 x 0,y 0,z 0 ", B, 1, 1.5, Text"P 1 x 1,y 1,z 1 ", A Text"n ", A NORMAL,, 0, Text"v BA NORMAL ",,, 0, DrawVector3DB, A NORMAL, HeadLength 0.05, HeadAngle DisplayFunction Identity, DrawVector3DA, A NORMAL, HeadAngle 0.05, ShaftColor Red, DisplayFunction Identi Ticks False, Boxed True, ViewPoint 3, 3, 5, DisplayFunc

29 Line Segment between two points In[88]:= Needs"VisualLA`" rt_ 1t6, 6, 5t7, 1, ; A0 ParametricPlot3DFlattenrt, RGBColor0, 0, 1, t, 5 DisplayFunction Identity; A1 ParametricPlot3DFlattenrt, RGBColor0, 0, 1, t, 1, DisplayFunction Identity; B ParametricPlot3DFlattenrt, RGBColor0, 1, 0, t, 0, 1 PTS PointPlot3Dr0, r1, DisplayFunction Identity; LBL Text"r 06,6,5", r0, 1.5, 1, Text"r 17,1, ShowGraphics3DLBL, A0, B, A1, PTS, DisplayFunction $Displ PlotRange 10, 10, 10, 10, 10, 10, Ticks False, A Boxed True, ViewPoint.49, 1.548, 1.775;

30 Intersection of planes In[414]:= Planes P1x_, y_ xy; Px_, y_ x5y1; F Plot3DP1x, y, x, 3, 3, y, 3, 3, DisplayFunction Ide G Plot3DPx, y, x, 3, 3, y, 3, 3, DisplayFunction Ide Vectors N1 1, 1, 1; N, 5, 1; N1xN CrossN1, N; PN1 DrawVector3D13, 13, P113, 13, 13, 13, DisplayFunction Identity; PN DrawVector3D13, 13, P13, 13, 13, 13, DisplayFunction Identity; DIRVEC DrawVector3D13, 13, P13, 13, 13, 1 DisplayFunction Identity; Graph ShowF, G, PN1, PN, DisplayFunction $DisplayFunction, Ticks ShowF, G, PN1, PN, DIRVEC, DisplayFunction $DisplayFunctio ViewPoint, 1, ;