Solve the following equation for the vector \( \vec{p} ; \vec{p} \times \vec{a} \) \( +(\vec{p} ...

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Solve the following equation for the vector \( \vec{p} ; \vec{p} \times \vec{a} \) \( +(\vec{p} \cdot \vec{b}) \vec{c}=\vec{b} \times \vec{c} \) where \( \vec{a}, \vec{b}, \vec{c} \) are non-zero non-coplanar vectors and \( \vec{a} \) is neither perpendicular to \( \vec{b} \) nor to \( \vec{c} \), hence show that \( \left(\vec{p} \times \vec{a}+\frac{[\vec{a} \vec{b} \vec{c}]}{\vec{a} \cdot \vec{c}} \vec{c}\right) \) is perpendicular to \( \vec{b}-\vec{c} \).
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