Let → a = ^ i + ^ j + 2 ^ k and → b = - ^ i + 2 ^ j + 3 ^ k . Then the vector product ( → a + → b ) × (( → a × (( → a - → b ) × → b )) × → b ) is equal to :

Correct answer: B. 7(34 ^ i - 5 ^ j + 3 ^ k ) → a = ^ i + ^ j + 2 ^ k → b = - ^ i + 2 ^ j + 3 ^ k → a + → b = 3 ^ j + 5 ^ k ; → a · → b = - 1 + 2 + 6 = 7 Now (( → a × (( → a - → b ) × → b )) × → b ) = (( → a × ( → a × → b - → b × → b )) × → b ) = ( → a × ( → a × → b - 0)) × → b = ( → a × ( → a × → b )) × → b = (( → a · → b ) → a - ( → a · → a ) → b ) × → b = ( → a · → b )( → a × → b ) - ( → a · → a )( → b × → b ) = ( → a · → b )( → a × → b ) Here, → a × → b = | cases ^ i & ^ j & ^ k 1 & 1 & 2 - 1 & 2 & 3 cases | = - ^ i - 5 ^ j + 3 ^ k ∴ ( → a · → b )( → a × → b ) = 7(- ^ i - 5 ^ j + 3 ^ k ) So, ( → a + → b ) × (7(- ^ i - 5 ^ j + 3 ^ k )) 7(0 ^ i + 3 ^ j + 5 ^ k ) × (- ^ i - 5 ^ j + 3 ^ k ) = 7| cases ^ i & ^ j & ^ k 0 & 3 & 5 - 1 & - 5 & 3 cases | = 7[34 ^ i - (5) ^ j + (3 ^ k )] = 7[34 ^ i - 5 ^ j + 3 ^ k ]

JEE Main 202127 JulyMorning Shift

Exam: JEE Main
Year: 2021
Date: 27 July
Shift: Morning Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2021 · 27 July · Morning Shift · Mathematics · Vector Algebra

Let → a = ^ i + ^ j + 2 ^ k and → b = - ^ i + 2 ^ j + 3 ^ k . Then...

Let

\overset{\to}{a} = \overset{^}{i} + \overset{^}{j} + 2\overset{^}{k} and

\overset{\to}{b} = - \overset{^}{i} + 2\overset{^}{j} + 3\overset{^}{k} .

Then the vector product \left(\overset{\to}{a} + \overset{\to}{b}\right) \times \left(\left(\overset{\to}{a} \times \left(\left(\overset{\to}{a} - \overset{\to}{b}\right) \times \overset{\to}{b}\right)\right) \times \overset{\to}{b}\right)

is equal to :

Options

A $5\left(34\overset{^}{i} - 5\overset{^}{j} + 3\overset{^}{k}\right)$
B $7\left(34\overset{^}{i} - 5\overset{^}{j} + 3\overset{^}{k}\right)$
C $7\left(30\overset{^}{i} - 5\overset{^}{j} + 7\overset{^}{k}\right)$
D $5\left(30\overset{^}{i} - 5\overset{^}{j} + 7\overset{^}{k}\right)$

Correct answer

B. $7\left(34\overset{^}{i} - 5\overset{^}{j} + 3\overset{^}{k}\right)$

Step-by-step solution

$\overset{\to}{a} = \overset{^}{i} + \overset{^}{j} + 2\overset{^}{k}$

$\overset{\to}{b} = - \overset{^}{i} + 2\overset{^}{j} + 3\overset{^}{k}$

$\overset{\to}{a} + \overset{\to}{b} = 3\overset{^}{j} + 5\overset{^}{k}; \overset{\to}{a} \cdot \overset{\to}{b} = - 1 + 2 + 6 = 7$

Now \left(\left(\overset{\to}{a} \times \left(\left(\overset{\to}{a} - \overset{\to}{b}\right) \times \overset{\to}{b}\right)\right) \times \overset{\to}{b}\right) $$ = \left(\left(\overset{\to}{a} \times \left(\overset{\to}{a} \times \overset{\to}{b} - \overset{\to}{b} \times \overset{\to}{b}\right)\right) \times \overset{\to}{b}\right) $$ = \left(\overset{\to}{a} \times \left(\overset{\to}{a} \times \overset{\to}{b} - 0\right)\right) \times \overset{\to}{b} $$ = \left(\overset{\to}{a} \times \left(\overset{\to}{a} \times \overset{\to}{b}\right)\right) \times \overset{\to}{b} $$ = \left(\left(\overset{\to}{a} \cdot \overset{\to}{b}\right)\overset{\to}{a} - \left(\overset{\to}{a} \cdot \overset{\to}{a}\right)\overset{\to}{b}\right) \times \overset{\to}{b} $$ = \left(\overset{\to}{a} \cdot \overset{\to}{b}\right)\left(\overset{\to}{a} \times \overset{\to}{b}\right) - \left(\overset{\to}{a} \cdot \overset{\to}{a}\right)\left(\overset{\to}{b} \times \overset{\to}{b}\right) $$ = \left(\overset{\to}{a} \cdot \overset{\to}{b}\right)\left(\overset{\to}{a} \times \overset{\to}{b}\right) $Here,$ \overset{\to}{a} \times \overset{\to}{b} = \left| $\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ 1 & 1 & 2 \\ - 1 & 2 & 3\end{cases}$ \right| = - \overset{^}{i} - 5\overset{^}{j} + 3\overset{^}{k} $$ ∴ \left(\overset{\to}{a} \cdot \overset{\to}{b}\right)\left(\overset{\to}{a} \times \overset{\to}{b}\right) = 7\left(- \overset{^}{i} - 5\overset{^}{j} + 3\overset{^}{k}\right) $So, \left(\overset{\to}{a} + \overset{\to}{b}\right) \times \left(7\left(- \overset{^}{i} - 5\overset{^}{j} + 3\overset{^}{k}\right)\right)$

$7\left(0\overset{^}{i} + 3\overset{^}{j} + 5\overset{^}{k}\right) \times \left(- \overset{^}{i} - 5\overset{^}{j} + 3\overset{^}{k}\right)$

$= 7\left|$ \begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ 0 & 3 & 5 \\ - 1 & - 5 & 3\end{cases} $\right|$

$= 7\left[34\overset{^}{i} - \left(5\right)\overset{^}{j} + \left(3\overset{^}{k}\right)\right]$

$= 7\left[34\overset{^}{i} - 5\overset{^}{j} + 3\overset{^}{k}\right]$

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