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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Lines and Planes
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
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
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.
Find the equation of the plane through the points (-1, 2, -4), (2, -3, 4) and (2, 1, -3).
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.
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
The line through P and Q is given by
x(t) = (1 – t)p + tq
Note that x(0) = p and x(1) = q.