1 / 12

Geometry of R 2 and R 3

Geometry of R 2 and R 3. Lines and Planes. Point-Normal Form for a Plane. Let P be a point in R 3 and n a nonzero vector. Then the plane that contains P that is normal to the vector n is given by the equation n . ( x - p ) = 0. Standard Form for a Plane.

baker-lynch
Download Presentation

Geometry of R 2 and R 3

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. Geometry of R2 and R3 Lines and Planes

  2. Point-Normal Form for a Plane Let P be a point in R3 and n a nonzero vector. Then the plane that contains P that is normal to the vector n is given by the equation n.(x - p) = 0

  3. Standard Form for a Plane Let n = (a, b, c) and x = (x, y, z) in the point-normal form we get the standard form of the equation of the plane ax + by + cz = d

  4. Example • Find the equation of the plane through the point (1, 2, 3) with normal n = (-3, 0, 1). • Convert the equation 4x – 3y + 6z = 12 of the plane to point-normal form.

  5. Plane Determined by Three Points If P, Q, and R are three non-collinear points in a plane, then n = (q – p) x (r – p) and the equation is again n.(x - p) = 0.

  6. Example Find the equation of the plane through the points (-1, 2, -4), (2, -3, 4) and (2, 1, -3).

  7. Point-Parallel Form for a Line Let P be a point and v a nonzero vector in R3. Then the p + tv is parallel and equal in length to the vector tv. Then the endpoint of p + tv must lie on the line determined by P and the endpoint of p + v. So for any point X on the line through P and parallel to v is the end point of a vector of the form p + tv. Thus x(t) = p + tv the line through P and parallel to the v.

  8. Example • Find the point-parallel form of the line through the point (2, -1, 3), parallel to v = (-1, 4, 1). • Find the point-parallel form of the line through (-2, 3, 0) and (3,-1,-2). • Find the point-parallel form of the line through (-3,1) and parallel to v = (4, -3).

  9. Parametric Equations for a Line Recall: x = p + tv the line through P and parallel to the v. Let x = (x, y, z), p = (p1, p2, p3) and v = (v1, v2, v3). Then the parametric equation of the above line is x(t) = p1 + tv1; y(t) = p2 + tv2; z(t) = p3 + tv3

  10. Example • Find the parametric form of the line through (2, -1, 3), parallel to v = (-1, 4, 1). • Find the parametric form of the line through (2, 4, 5), perpendicular to the plane 5x – 5y – 10z = 2

  11. Two-Point Form for a Line The line through P and Q is given by x(t) = (1 – t)p + tq Note that x(0) = p and x(1) = q.

  12. Homework 1.3

More Related