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    Plane Equation, Normal, Distance

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    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]|
    |unv|

    (Please see attachment for proper formatting).

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    https://brainmass.com/math/vector-calculus/plane-equation-normal-distance-539993

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    a.
    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 Summary

    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.

    $2.19