Let → a = - 5 ^ i + ^ j - 3 ^ k , → b = ^ i + 2 ^ j - 4 ^ k and → c = ((( → a × → b ) × ^ i ) × ^ i ) × ^ i . Then → c ⋅ (- ^ i + ^ j + ^ k ) is equal to

Correct answer: A. - 12 Given: → a = - 5 ^ i + j - 3 ^ k and → b = ^ i + 2 ^ j - 4 ^ k ⇒ ( → a × → b ) × ^ i = ( → a ⋅ ^ i ) → b - ( → b ⋅ ^ i ) → a ⇒ ( → a × → b ) × ^ i = - 5 → b - → a ⇒ ( → a × → b ) × ^ i = (((- 5 → b - → a ) × ^ i ) × ^ i ) ⇒ ( → a × → b ) × ^ i = ((- 11 ^ j + 23 ^ k ) × ^ i ) × ^ i ⇒ ( → a × → b ) × ^ i = (11 ^ k + 23 ^ j ) × ^ i ⇒ ( → a × → b ) × ^ i = (11 ^ j - 23 ^ k ) ⇒ → c . (- ^ i + ^ j + ^ k ) = 11 - 23 ⇒ → c . (- ^ i + ^ j + ^ k ) = - 12

JEE Main 20241 FebruaryMorning Shift

Exam: JEE Main
Year: 2024
Date: 1 February
Shift: Morning Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2024 · 1 February · Morning Shift · Mathematics · Vector Algebra

Let → a = - 5 ^ i + ^ j - 3 ^ k , → b = ^ i + 2 ^ j - 4 ^ k and → c...

Let

\overset{\to}{a} = - 5\overset{^}{i} + \overset{^}{j} - 3\overset{^}{k}, \overset{\to}{b} = \overset{^}{i} + 2\overset{^}{j} - 4\overset{^}{k}

and

\overset{\to}{c} = \left(\left(\left(\overset{\to}{a} \times \overset{\to}{b}\right) \times \overset{^}{i}\right) \times \overset{^}{i}\right) \times \overset{^}{i}

. Then

\overset{\to}{c} ⋅ \left(- \overset{^}{i} + \overset{^}{j} + \overset{^}{k}\right)

is equal to

Options

A $- 12$
B $- 10$
C $- 13$
D $- 15$

Correct answer

A. $- 12$

Step-by-step solution

Given:

$\overset{\to}{a} = - 5\overset{^}{i} + j - 3\overset{^}{k}$ and $\overset{\to}{b} = \overset{^}{i} + 2\overset{^}{j} - 4\overset{^}{k}$

$⇒ \left(\overset{\to}{a} \times \overset{\to}{b}\right) \times \overset{^}{i} = \left(\overset{\to}{a} ⋅ \overset{^}{i}\right)\overset{\to}{b} - \left(\overset{\to}{b} ⋅ \overset{^}{i}\right)\overset{\to}{a}$

$⇒ \left(\overset{\to}{a} \times \overset{\to}{b}\right) \times \overset{^}{i} = - 5\overset{\to}{b} - \overset{\to}{a}$

$⇒ \left(\overset{\to}{a} \times \overset{\to}{b}\right) \times \overset{^}{i} = \left(\left(\left(- 5\overset{\to}{b} - \overset{\to}{a}\right) \times \overset{^}{i}\right) \times \overset{^}{i}\right)$

$⇒ \left(\overset{\to}{a} \times \overset{\to}{b}\right) \times \overset{^}{i} = \left(\left(- 11\overset{^}{j} + 23\overset{^}{k}\right) \times \overset{^}{i}\right) \times \overset{^}{i}$

$⇒ \left(\overset{\to}{a} \times \overset{\to}{b}\right) \times \overset{^}{i} = \left(11\overset{^}{k} + 23\overset{^}{j}\right) \times \overset{^}{i}$

$⇒ \left(\overset{\to}{a} \times \overset{\to}{b}\right) \times \overset{^}{i} = \left(11\overset{^}{j} - 23\overset{^}{k}\right)$

$⇒ \overset{\to}{c} . \left(- \overset{^}{i} + \overset{^}{j} + \overset{^}{k}\right) = 11 - 23$

$⇒ \overset{\to}{c} . \left(- \overset{^}{i} + \overset{^}{j} + \overset{^}{k}\right) = - 12$

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