JEE MAIN - Mathematics (2021 - 1st September Evening Shift - No. 21)
Let $$\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k$$ and $$\overrightarrow b = \widehat i + 2\widehat j - \widehat k$$. Let a vector $$\overrightarrow v $$ be in the plane containing $$\overrightarrow a $$ and $$\overrightarrow b $$. If $$\overrightarrow v $$ is perpendicular to the vector $$3\widehat i + 2\widehat j - \widehat k$$ and its projection on $$\overrightarrow a $$ is 19 units, then $${\left| {2\overrightarrow v } \right|^2}$$ is equal to _____________.
Answer
1494
Explanation
$$\overrightarrow a = 2\widehat i - \widehat j + 2\widehat k$$
$$\overrightarrow b = \widehat i + 2\widehat j - \widehat k$$
$$\overrightarrow c = 3\widehat i + 2\widehat j - \widehat k$$
$$\overrightarrow v = x\overrightarrow a + y\overrightarrow b $$
$$\overrightarrow v \left( {3\widehat i + 2\widehat j - \widehat k} \right) = 0$$
$$\overrightarrow v .\widehat a = 19$$
$$\overrightarrow v = \lambda \overrightarrow c \times \left( {\overrightarrow a \times \overrightarrow b } \right)$$
$$\overrightarrow v = \lambda \left[ {\left( {\overrightarrow c .\overrightarrow b } \right)\overrightarrow a - \left( {\overrightarrow c .\overrightarrow a } \right)\overrightarrow b } \right]$$
$$ = \lambda [(3 + 4 + 1)\left( {2\widehat i - \widehat j + 2\widehat k} \right) - \left( {{{6 - 2 - 2} \over 2}} \right)\left( {\widehat i + 2\widehat j + \widehat k} \right)$$
$$ = \lambda [16\widehat i - 8\widehat j + 16\widehat k - 2\widehat i - 4\widehat j + 2\widehat k]$$
$$\overrightarrow v = \lambda \left[ {14\widehat i - 12\widehat j + 18\widehat k} \right]$$
$$\lambda [14\widehat i - 12\widehat j + 18\widehat k].{{\left( {2\widehat i - \widehat j + 2\widehat k} \right)} \over {\sqrt {4 + 1 + 4} }} = 19$$
$$\lambda {{[28 + 12 + 36]} \over 3} = 19$$
$$\lambda \left( {{{76} \over 3}} \right) = 19$$
$$4\lambda = 3 \Rightarrow \lambda = {3 \over 4}$$
$$|2{v^2}| = {\left| {2 \times {3 \over 4}(14\widehat i - 12\widehat j + 18\widehat k)} \right|^2}$$
$${9 \over 4} \times 4{\left( {7\widehat i - 6\widehat j + 9\widehat k} \right)^2}$$
$$ = 9(49 + 36 + 81)$$
$$ = 9(166)$$
$$ = 1494$$
$$\overrightarrow b = \widehat i + 2\widehat j - \widehat k$$
$$\overrightarrow c = 3\widehat i + 2\widehat j - \widehat k$$
$$\overrightarrow v = x\overrightarrow a + y\overrightarrow b $$
$$\overrightarrow v \left( {3\widehat i + 2\widehat j - \widehat k} \right) = 0$$
$$\overrightarrow v .\widehat a = 19$$
$$\overrightarrow v = \lambda \overrightarrow c \times \left( {\overrightarrow a \times \overrightarrow b } \right)$$
$$\overrightarrow v = \lambda \left[ {\left( {\overrightarrow c .\overrightarrow b } \right)\overrightarrow a - \left( {\overrightarrow c .\overrightarrow a } \right)\overrightarrow b } \right]$$
$$ = \lambda [(3 + 4 + 1)\left( {2\widehat i - \widehat j + 2\widehat k} \right) - \left( {{{6 - 2 - 2} \over 2}} \right)\left( {\widehat i + 2\widehat j + \widehat k} \right)$$
$$ = \lambda [16\widehat i - 8\widehat j + 16\widehat k - 2\widehat i - 4\widehat j + 2\widehat k]$$
$$\overrightarrow v = \lambda \left[ {14\widehat i - 12\widehat j + 18\widehat k} \right]$$
$$\lambda [14\widehat i - 12\widehat j + 18\widehat k].{{\left( {2\widehat i - \widehat j + 2\widehat k} \right)} \over {\sqrt {4 + 1 + 4} }} = 19$$
$$\lambda {{[28 + 12 + 36]} \over 3} = 19$$
$$\lambda \left( {{{76} \over 3}} \right) = 19$$
$$4\lambda = 3 \Rightarrow \lambda = {3 \over 4}$$
$$|2{v^2}| = {\left| {2 \times {3 \over 4}(14\widehat i - 12\widehat j + 18\widehat k)} \right|^2}$$
$${9 \over 4} \times 4{\left( {7\widehat i - 6\widehat j + 9\widehat k} \right)^2}$$
$$ = 9(49 + 36 + 81)$$
$$ = 9(166)$$
$$ = 1494$$
Comments (0)
