If `A( vec a),B( vec b)a n dC( vec c)` are three non-collinear points and origin does not lie in the plane of the points `A ,Ba n dC ,` then point `P( vec p)` in the plane of the ` A B C` such that vector ` vec O P` is `_|_` to planeof ` A B C` , show that ` vec O P=([ vec a vec b vec c]( vec axx vec b+ vec bxx vec c+ vec cxx vec a))/(4^2),w h e r e` is the area of the ` A B Cdot`

Explanation

To solve the problem step by step, we need to show that the vector \( \vec{OP} \) can be expressed as:

\[ \vec{OP} = \frac{[\vec{a}, \vec{b}, \vec{c}](\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a})}{4^2} \]

where \( [\vec{a}, \vec{b}, \vec{c}] \) represents the area of triangle \( ABC \).

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