Let a =a_ 1 i +a_ 2 j +a_ 3 k , b = b _ 1 i + b _ 2 j + b _ 3 k and c = c _ 1 i + c _ 2 j + c _ 3 k be three non–zero vectors such that c is a unit vector perpendicular to both a and b . If the angle between a and b is π/6, then | array lll a _ 1 & a _ 2 & a _ 3 ~b _ 1 & ~b _ 2 & ~b _ 3 c _ 1 & c _ 2 & c _ 3 array |^ 2 is equal to :

Let a =a_ 1 i +a_ 2 j +a_ 3 k , b = b _ 1 i + b _ 2 j + b _ 3 k and c = c _ 1 i + c _ 2 j…

Mathematics · JEE Advanced · NTA Exams — Three Dimensional Geometry

Let \(\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k},\) \(\overrightarrow{\mathrm{b}}=\mathrm{b}_{1} \hat{\mathrm{i}}+\mathrm{b}_{2} \hat{\mathrm{j}}+\mathrm{b}_{3} \hat{\mathrm{k}}\) and
\(\overrightarrow{\mathrm{c}}=\mathrm{c}_{1} \hat{\mathrm{i}}+\mathrm{c}_{2} \hat{\mathrm{j}}+\mathrm{c}_{3} \hat{\mathrm{k}}\) be three non–zero vectors such that \(\stackrel{\rightharpoonup}{\mathrm{c}}\) is a unit vector perpendicular to both \(\overrightarrow{\mathrm{a}} \text { and } \overrightarrow{\mathrm{b}}\). If the angle between \(\overrightarrow{\mathrm{a}} \text { and } \overrightarrow{\mathrm{b}}\) is π/6, then \[\left|\begin{array}{lll} \mathrm{a}_{1} & \mathrm{a}_{2} & \mathrm{a}_{3} \\ \mathrm{~b}_{1} & \mathrm{~b}_{2} & \mathrm{~b}_{3} \\ \mathrm{c}_{1} & \mathrm{c}_{2} & \mathrm{c}_{3} \end{array}\right|^{2}\] is equal to :

0
1
\(\frac{1}{4}\left(\mathrm{a}_{1}^{2}+\mathrm{a}_{2}^{2}+\mathrm{a}_{3}^{2}\right)\left(\mathrm{b}_{1}^{2}+\mathrm{b}_{2}^{2}+\mathrm{b}_{3}^{2}\right)\)
none of these

Answer

(C) 1 4 ( a _ 1 ^ 2 + a _ 2 ^ 2 + a _ 3 ^ 2 ) ( b _ 1 ^ 2 + b _ 2 ^ 2 + b _ 3 ^ 2 )

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Let a =a_ 1 i +a_ 2 j +a_ 3 k , b = b _ 1 i + b _ 2 j + b _ 3 k and c = c _ 1 i + c _ 2 j…

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