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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).

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

#### Solution Preview

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.

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