1 / 18

The parametric equations of a line

11.5 Lines and Planes in Space For an animation of this topic visit: http://www.math.umn.edu/~nykamp/m2374/readings/lineparam/. The parametric equations of a line. Symmetric equations of the line. Example 1.

juliet-sanford
Download Presentation

The parametric equations of a line

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 11.5 Lines and Planes in SpaceFor an animation of this topic visit:http://www.math.umn.edu/~nykamp/m2374/readings/lineparam/

  2. The parametric equations of a line

  3. Symmetric equations of the line

  4. Example 1 • Find the parametric and symmetric equations of the line L that passes through (1,-2,4) and is parallel to v = <2,4,-4>

  5. Example 1 solution Solution to find a set of parametric equations of the line, use the coordinates x1 =1, y1=-2, z1 = 4 and direction numbers a=2, b = 4 and c=-4 x= 1+2t, y = -2+4t, z=4-4t (parametric equations) Because a,b and c are all nonzero, a set of symmetric equations is Find parametric and symmetric equations of line L that passes through the point (1,-2,4 ) and is parallel to v = <2,4,-4>

  6. Example 2 • Find a set of parametric equations of the line that passes through the points (-2,1,0) and (1,3,5), Graph the equation (next slide shows axis).

  7. Example 2 solution

  8. Problem 26Determine if any of the following lines are parallel. L1: (x-8)/4 = (y-5)/-2 = (z+9)/3 L2: (x+7)/2 = (y-4)/1 = (z+6)/5 L3: (x+4)/-8 = (y-1)/4 = (z+18)/-6 L4: (x-2)/-2 = (y+3)/1 = (z-4)/1.5

  9. Problem 28 Determine whether lines intersect, and if so find the point of intersection and the angle of intersection. x = -3t+1, y = 4t + 1, z = 2t+4 x=3s +1, y = 4s +1, z = -s +1

  10. Solution 28 x = -3t+1, y = 4t + 1, z = 2t+4 x=3s +1, y = 4s +1, z = -s +1 Set x y and z equations equal -3t+1 =3s +1 4s +1 = 4t + 1 -s +1 = 2t+4 From the first one we get s=-t When s=-t is plugged into the second we get t=1/3 When plugged into the third equation we get t = -3 Hence the lines do not intersect

  11. Problem 30 Determine whether lines intersect, and if so find the point of intersection and the angle of intersection. (x-2)/-3 = (y-2)/6 = z-3 (x-3)/2=y+5=(z+2)/4

  12. Hint on Problem 30 (x-2)/-3 = (y-2)/6 = z-3 (x-3)/2=y+5=(z+2)/4 Write the equations in parametric form Then solve as per #28 These do intersect. Use the dot product of the direction vectors to find the angle between the two lines <-3,6,1> ∙ <2,1,4>

  13. Example 3 • Find the equation of a plane containing points (2,1,1), (0,4,1) and (-2,1,4)

  14. Example 3 Solution

  15. Drawing a plane

More Related