Consider the plane P through points (1, 2;,2), (2,1, 2), and (3,4,0). (a) Find a

Question

Consider the plane P through points (1, 2;,2), (2,1, 2), and (3,4,0). (a) Find a parametrization for the plane P. (b) Find a linear (Cartesian) equation for the plane P. Perform any computations by hand. (c) Plot your answers to (a) and (b) to check that they produce the same plane. (d) Give the Cartesian equation for any plane that intersects P at an angle of =3.

Explanation / Answer

eqn of a plane = ax + by + cz + d = 0 ---> x + (b/a)y + (c/a)z + (d/a) = 0 ---> x + py + qz + r = 0 putting each point: 1 + 2p + 2q + r = 0 .......(i) 2 + p + 2q + r = 0 ........(ii) 3 + 4p + r = 0 ............(iii) solving, we get: p = 1, q = 2 and r = -7 so, eqn: x + y + 2z = 7

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