JEE MAIN - Mathematics (2020 - 4th September Evening Slot - No. 11)
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____
$${\left| {\widehat i \times \left( {\overrightarrow a \times \widehat i} \right)} \right|^2} + {\left| {\widehat j \times \left( {\overrightarrow a \times \widehat j} \right)} \right|^2} + {\left| {\widehat k \times \left( {\overrightarrow a \times \widehat k} \right)} \right|^2}$$ is equal to____
Answer
18
Explanation
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
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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