question_answer

Let \[\overrightarrow{a},\] \[\overrightarrow{b}\] and \[\overrightarrow{c}\] be non-coplanar unit vectors, equally inclined to one another at an angle\[\theta \]. If \[\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}=p\overrightarrow{a}+q\overrightarrow{b}+r\overrightarrow{c},\] find the scalars p, q and r in terms of\[\theta \].

A)  \[p=\frac{1}{\sqrt{1+2\cos \theta }},\]\[q=\frac{-2\cos \theta }{\sqrt{1+2\cos \theta }},\]\[r=\frac{1}{\sqrt{1+2\cos \theta }}\]

B)  \[p=\frac{-2\cos \theta }{\sqrt{1+2\cos \theta }},\]\[q=\frac{1}{\sqrt{1+2\cos \theta }},\]\[r=\frac{1}{\sqrt{1+2\cos \theta }}\]

C)  \[p=\frac{1}{\sqrt{1+2\cos \theta }},\]\[q=\frac{1}{\sqrt{1+2\cos \theta }},\]\[r=\frac{-2\cos \theta }{\sqrt{1+2\cos \theta }}\]

D)  \[p=\frac{1}{\sqrt{1+2\cos \theta }},\]\[q=\frac{-2\cos \theta }{\sqrt{1+2\cos \theta }},\]\[r=\frac{-1}{\sqrt{1+2\cos \theta }}\]

E)  None of these