geometry of r 2 and r 3
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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.

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geometry of r 2 and r 3

Geometry of R2 and R3

Lines and Planes

point normal form for a plane
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

standard form for a plane
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
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.
plane determined by three points
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.

example1
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
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.

example2
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).
parametric equations for a line
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

example3
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
two point form for a line
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.

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