Geometry of R 2 and R 3 Lines and Planes. Point-Normal Form for a Plane R 3 Let P be a point in R 3 and n a nonzero vector. Then the plane that contains.

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Presentation transcript:

Geometry of R 2 and R 3 Lines and Planes

Point-Normal Form for a Plane R 3 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 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

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

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.

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

Point-Parallel Form for a Line R 3 Let P be a point and v a nonzero vector in R 3. 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.

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

Parametric Equations for a Line Recall: x = p + tv the line through P and parallel to the v. Let x = (x, y, z), p = (p 1, p 2, p 3 ) and v = (v 1, v 2, v 3 ). Then the parametric equation of the above line is x(t) = p 1 + tv 1 ; y(t) = p 2 + tv 2 ; z(t) = p 3 + tv 3

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

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.

Homework 1.3