Lines Vector Parametrizations (13.5.1) r(t)=ro +id, t real (13.5.2) r(t)=(xo+td)i+(yo +td2)j+(zo+td3)k Figure 13.5.2 Figure 13.5.3 Main Meny C
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Lines Vector Parametrizations
Example:Parametrize the line that passes through the point P(1-1,2) and has direction vector 2i-3j+k Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example: Parametrize the line that passes through the point P (1,-1,2) and has direction vector 2i-3j+k
Lines Scalar Parametric Equations (13.5.3) x(t)=xo+dt, y(t)=0+d24, z(t)=zo +d3t Symmetric Form (13.5.4) x-0=y-地=2-0 d d3 Main Meny C 007e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Lines Scalar Parametric Equations Symmetric Form
Example:Write a parametric equations for the line that passes through the point P(1,-1,2)and has direction vector 2i-3j+k Main Meny c8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example: Write a parametric equations for the line that passes through the point P (1,-1,2) and has direction vector 2i-3j+k
Intersecting Lines,Parallel Lines Two distinct lines r(t)=ro+id,2 R(u)=Ro+uD intersect iff there are numbers t and u at which r()=R(u) Example:Find the point at which the lines 1:(t)=(i-6j+2k)+ti+2j+k), 12:R(w=(4j+k)+u(2i+j+2K) Intersect. Main Menu
Main Menu Intersecting Lines, Parallel Lines Two distinct lines l1 : r(t) = r0 + td, l2 : R(u) = R0 + uD intersect iff there are numbers t and u at which r(t) = R(u). Example: Find the point at which the lines Intersect. : ( ) (4 ) (2 2 ) : ( ) ( 6 2 ) ( 2 ), 2 1 l R u j k u i j k l r t i j k t i j k = + + + + = − + + + +
Example:(1)Find the angle between the lines 1:r(t)=(i-6j+2k)+ti+2j+k),1,:R)=(4j+k)+u(2i+j+2k) (2)Find the parametrization for the line that passes through their intersection and is perpendicular to both/,and 12 Main Menu
Main Menu Example: (1) Find the angle between the lines (2) Find the parametrization for the line that passes through their intersection and is perpendicular to both l1 and l2. : ( ) ( 6 2 ) ( 2 ), : ( ) (4 ) (2 2 ) 1 2 l r t i j k t i j k l R u j k u i j k = − + + + + = + + + +
Lines Distance from a Point to a Line Let Po be a point on and let d be a direction vector for l.With Po and O as shown in the figure,you can see that d(P,1)=d(P.Q)=PPsin0 Figure 13.5.8 (13.5.6) dB,)= IPoE×d d Main Meny w
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Lines Distance from a Point to a Line Let P0 be a point on l and let d be a direction vector for l. With P0 and Q as shown in the figure, you can see that ( 1 1 0 ) ( ) 1 d P l d P Q P P , , sin = =
Planes Scalar Equation of a Plane N=Ai++Ck P(to.Vo.zo) Figure 13.6.1 (13.6.1) Ax-x0)+B0-%)+C(2-20)=0. Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Planes Scalar Equation of a Plane
Planes Vector Equation of a Plane We can write the equation of a plane entirely in vector notation.Set N=Ai+j+Ck,0=O丽=(x,%5),r=00=(x,y,） Since ro=xoi+yaj+zok and r=xi+yj+zk (13.6.1)can be written (13.6.2) N(r-ro)=0. Figure13.6.3 Main Meny 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Planes Vector Equation of a Plane We can write the equation of a plane entirely in vector notation. Set Since r0 = x0 i + y0 j + z0k and r = xi + yj + zk, (13.6.1) can be written N i j k r r = + + = = = = A B C OP x y z OQ x y z , , , , , , 0 0 0 0 ( ) ( )
Planes Unit Normals If N is normal to a given plane,then all other normals to that plane are parallel to N and hence scalar multiples of N.In particular there are two normals of length 1: amnd-N N Intersecting Planes (13.6.3) cos0=luN,·,l Figure 13.6.4 Main Meny c墙8的整
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Planes Intersecting Planes Unit Normals If N is normal to a given plane, then all other normals to that plane are parallel to N and hence scalar multiples of N. In particular there are two normals of length 1: = − = and N N N N u u N N