The position vectors of three points A, B and C are \vec{a}, \vec{b} and \vec{c} respectively, where \vec{c}=(\cos^{2}\theta)\vec{a}+(\sin^{2}\theta)\vec{b}. What is (\vec{a}\times\vec{b})+(\vec{b}\times\vec{c})+(\vec{c}\times\vec{a}) equal to?

Explanation

Since the sum of the scalar coefficients \cos^2\theta + \sin^2\theta = 1, the vector \vec{c} lies on the line joining \vec{a} and \vec{b}. This implies the points A, B, and C are collinear. The expression (\vec{a}\times\vec{b})+(\vec{b}\times\vec{c})+(\vec{c}\times\vec{a}) represents twice the vector area of triangle ABC. Since they are collinear, the area is \vec{0}.

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