a) Justify the vector equation for a plane in the form r = a + lambda*u + mu*v, where a, u, v are non-coplanar vectors and lambda, mu are arbitrary scalars.
b) What is the normal vector to the plane?
c) Show that the shortest distance of a point with position vector p from the above plane is given by:
|[p-a, u, v]|
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Let us assume a general point P in the plane with position vector: r_vec
Point A in the plan has position vector: a_vec
There are two non-collinear vectors u_vec and v_vec in the plane.
Hence, a_vec, u_vec and v_vec are non-coplanar.
vector AP = vector OP - vector OA = r_vec - a_vec
In terms of u_vec and ...
Solution for an equation of a plane in terms of non-coplanar vectors, followed by normal to the plane and distance of the plane from the given point.