JEE Advance - Mathematics (1986 - No. 2)
Let $$\overrightarrow a = {a_1}i + {a_2}j + {a_3}k,\,\,\,\overrightarrow b = {b_1}i + {b_2}j + {b_3}k$$ and $$\overrightarrow c = {c_1}i + {c_2}j + {c_3}k$$ be three non-zero vectors such that $$\overrightarrow c $$ is a unit vector perpendicular to both the vectors $$\overrightarrow a $$ and $$\overrightarrow b .$$ If the angle between $$\overrightarrow a $$ and $$\overrightarrow b $$ is $${\pi \over 6},$$ then
$${\left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{b_1}} & {{b_2}} & {{b_3}} \cr {{c_1}} & {{c_2}} & {{c_3}} \cr } } \right|^2}$$ is equal to
$${\left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{b_1}} & {{b_2}} & {{b_3}} \cr {{c_1}} & {{c_2}} & {{c_3}} \cr } } \right|^2}$$ is equal to
$$0$$
$$1$$
$${1 \over 4}\left( {a_1^2 + a_2^2 + a_2^3} \right)\left( {b_1^2 + b_2^2 + b_3^2} \right)$$
$${3 \over 4}\left( {a_1^2 + a_2^2 + a_3^2} \right)\left( {b_1^2 + b_2^2 + b_3^2} \right)\left( {c_1^2 + c_2^2 + c_3^2} \right)$$
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