1. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Lines Vector Parametrizations

2. 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

3. 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

4. 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

5. Main Menu Intersecting Lines, Parallel Lines Two distinct lines l 1 : r(t) = r 0 + td, l 2 : R(u) = R 0 + uD intersect iff there are numbers t and u at which r(t) = R(u). Example: Find the point at which the lines Intersect.

6. 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 l 1 and l 2.

7. 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 P 0 be a point on l and let d be a direction vector for l. With P 0 and Q as shown in the figure, you can see that

8. 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

9. 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 r 0 = x 0 i + y 0 j + z 0 k and r = xi + yj + zk, (13.6.1) can be written

10. 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:

11. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.

12. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Example: Show that the plane are non-parallel and find a scalar parametric equations for the line formed by the two intersecting planes.

13. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Planes The Plane Determined by Three Noncollinear Points

14. Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Planes The Distance from a Point to a Plane