A) \[\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}|}\]
B) \[\frac{\,[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{a}\times \mathbf{b}+\mathbf{b}\times \mathbf{c}|}\]
C) \[\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}|}\]
D) \[\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{c}\times \mathbf{a}+\mathbf{a}\times \mathbf{b}|}\]
Correct Answer: C
Solution :
The given plane passes through \[\mathbf{a}\] and is parallel to the vectors \[\mathbf{b}-\mathbf{a}\] and \[\mathbf{c}\]. So it is normal to \[(\mathbf{b}-\mathbf{a})\times \mathbf{c}\]. Hence, its equation is \[(\mathbf{r}-\mathbf{a}).((\mathbf{b}-\mathbf{a})\times \mathbf{c})=0\] or \[\mathbf{r}.(\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a})=[\mathbf{a}\,\mathbf{b}\,\mathbf{c}\,]\] The length of the perpendicular from the origin to this plane is \[\frac{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}{|\mathbf{b}\times \mathbf{c}+\mathbf{c}\times \mathbf{a}|}\].
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