A) \[\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,=\overset{\to }{\mathop{\mathbf{b}}}\,\cdot \overset{\to }{\mathop{\mathbf{c}}}\,=0\]
B) \[\overset{\to }{\mathop{\mathbf{b}}}\,\cdot \overset{\to }{\mathop{\mathbf{c}}}\,=\overset{\to }{\mathop{\mathbf{c}}}\,\cdot \overset{\to }{\mathop{\mathbf{a}}}\,=0\]
C) \[\overset{\to }{\mathop{\mathbf{c}}}\,\cdot \overset{\to }{\mathop{\mathbf{a}}}\,=\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,=0\]
D) \[\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,=\overset{\to }{\mathop{\mathbf{b}}}\,\cdot \overset{\to }{\mathop{\mathbf{c}}}\,=\overset{\to }{\mathop{\mathbf{c}}}\,\cdot \overset{\to }{\mathop{\mathbf{a}}}\,=0\]
Correct Answer: D
Solution :
We have, \[|(\overset{\to }{\mathop{\mathbf{a}}}\,\times \overset{\to }{\mathop{\mathbf{b}}}\,)\cdot \overset{\to }{\mathop{\mathbf{c}}}\,|=|\overset{\to }{\mathop{\mathbf{a}}}\,||\overset{\to }{\mathop{\mathbf{b}}}\,||\overset{\to }{\mathop{\mathbf{c}}}\,|\] \[\Rightarrow \] \[||\overset{\to }{\mathop{\mathbf{a}}}\,||\overset{\to }{\mathop{\mathbf{b}}}\,|\sin \theta \,\widehat{\mathbf{n}}\cdot \overset{\to }{\mathop{\mathbf{c}}}\,|=|\overset{\to }{\mathop{\mathbf{a}}}\,||\overset{\to }{\mathop{\mathbf{b}}}\,||\overset{\to }{\mathop{\mathbf{c}}}\,|\] Where \[\widehat{\mathbf{n}}\] is a unit vector\[\bot \]to\[\overset{\to }{\mathop{\mathbf{a}}}\,\times \overset{\to }{\mathop{\mathbf{b}}}\,\] \[\Rightarrow \]\[|\overset{\to }{\mathop{\mathbf{a}}}\,||\overset{\to }{\mathop{\mathbf{b}}}\,||\overset{\to }{\mathop{\mathbf{c}}}\,|\sin \theta \cos \alpha |=|\overset{\to }{\mathop{\mathbf{a}}}\,||\overset{\to }{\mathop{\mathbf{b}}}\,||\overset{\to }{\mathop{\mathbf{c}}}\,|\] \[\Rightarrow \] \[|\sin \theta ||\cos \alpha |=1\] \[\Rightarrow \] \[\theta =\frac{\pi }{2}\]and\[\alpha =0\] \[\Rightarrow \] \[\overset{\to }{\mathop{\mathbf{a}}}\,\bot \overset{\to }{\mathop{\mathbf{b}}}\,\]and\[\overset{\to }{\mathop{\mathbf{c}}}\,||\widehat{\mathbf{n}}\] \[\Rightarrow \] \[\overset{\to }{\mathop{\mathbf{a}}}\,\bot \overset{\to }{\mathop{\mathbf{b}}}\,\]and\[\overset{\to }{\mathop{\mathbf{c}}}\,\]to both\[\overset{\to }{\mathop{\mathbf{a}}}\,\]and\[\overset{\to }{\mathop{\mathbf{b}}}\,\] \[\Rightarrow \] \[\overset{\to }{\mathop{\mathbf{a}}}\,,\,\,\overset{\to }{\mathop{\mathbf{b}}}\,\]and\[\overset{\to }{\mathop{\mathbf{c}}}\,\]are mutually perpendicular. \[\Rightarrow \] \[\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,=\overset{\to }{\mathop{\mathbf{b}}}\,\cdot \overset{\to }{\mathop{\mathbf{c}}}\,=\overset{\to }{\mathop{\mathbf{c}}}\,\cdot \overset{\to }{\mathop{\mathbf{a}}}\,=0\]
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