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Q. If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit coplanar vectors, then the scalar triple product $[(2 \vec{a}-\vec{b})(2 \vec{b}-\vec{c})(2 \vec{c}-\vec{a})]$ is

Vector Algebra

Solution:

Since $\vec{a}, \vec{b}$ and $\vec{c}$ are unit coplanar vectors, $2 \vec{a}-\vec{b}$, $2 \vec{b}-\vec{c}$ and $2 \vec{c}-\vec{a}$ are also coplanar vectors, being a linear combination of $\vec{a}, \vec{b}$ and $\vec{c}$.
Thus, $[(2 \vec{a}-\vec{b})(2 \vec{b}-\vec{c})(2 \vec{c}-\vec{a})]=0$