QJEEM20Madaea: JEE Main 2020 Physics - Class 11 JEEM20Madaea Solution: If , then the value of is equal to____

Question

Verified Solution • 2025–26 Session moderate

If

\overrightarrow a = 2\widehat i + \widehat j + 2\widehat k

, then the value of

{\left| {\widehat i \times \left( {\overrightarrow a \times \widehat i} \right)} \right|^2} + {\left| {\widehat j \times \left( {\overrightarrow a \times \widehat j} \right)} \right|^2} + {\left| {\widehat k \times \left( {\overrightarrow a \times \widehat k} \right)} \right|^2}

is equal to____

Accepted Answer

Let

\overrightarrow a = x\widehat i + y\widehat j + z\widehat k

Now

\widehat i \times \left( {\overrightarrow a \times \widehat i} \right) = \left( {\widehat i.\widehat i} \right)\overrightarrow a - \left( {\widehat i.\overrightarrow a } \right)\widehat i

=

y\widehat j + z\widehat k

Similarly

\widehat j \times \left( {\overrightarrow a \times \widehat j} \right) = x\widehat i + z\widehat k

\widehat k \times \left( {\overrightarrow a \times \widehat k} \right) = x\widehat i + y\widehat j

Now

{\left| {y\widehat j + z\widehat k} \right|^2} + {\left| {x\widehat i + z\widehat k} \right|^2} + {\left| {x\widehat i + y\widehat j} \right|^2}

=

2({x^2} + {y^2} + {z^2})

Given

\overrightarrow a = 2\widehat i + \widehat j + 2\widehat k

\therefore

x = 2, y = 1, z = 2

= 2(4 + 1 + 4) = 18

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