Let the vectors (2 + a + b) ^ i + (a + 2b + c) ^ j - (b + c) ^ k , (1 + b) ^ i + 2b ^ j - b ^ k and (2 + b) ^ i + 2b ^ j + (1 - b) ^ k , a, b, c ∈ R be co-planar. Then which of the following is true?

Correct answer: A. 2b = a + c If the vectors are co-planar, | cases a + b + 2 & a + 2b + c & - b - c b + 1 & 2b & - b b + 2 & 2b & 1 - b cases | = 0 Now R _ 1 ↔ R _ 1 - R _ 2 , R _ 3 ↔ R _ 3 - R _ 2 So | cases a + 1 & a + c & - c b + 1 & 2b & - b 1 & 0 & 1 cases | = 0 ⇒ ( a + 1 ) 2b - ( a + c ) ( 2b + 1 ) - c ( - 2b ) = 0 ⇒ 2ab + 2b - 2ab - a - 2bc - c + 2bc = 0 ⇒ 2b - a - c = 0 ⇒ 2b = a + c

JEE Main 202125 JulyMorning Shift

Exam: JEE Main
Year: 2021
Date: 25 July
Shift: Morning Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2021 · 25 July · Morning Shift · Mathematics · Vector Algebra

Let the vectors (2 + a + b) ^ i + (a + 2b + c) ^ j - (b + c) ^ k ,...

Let the vectors

\left(2 + a + b\right)\overset{^}{i} + \left(a + 2b + c\right)\overset{^}{j} - \left(b + c\right)\overset{^}{k}, \left(1 + b\right)\overset{^}{i} + 2b\overset{^}{j} - b\overset{^}{k} and \left(2 + b\right)\overset{^}{i} + 2b\overset{^}{j} + \left(1 - b\right)\overset{^}{k}, \forall a, b, c \in R

be co-planar. Then which of the following is true?

Options

A $2b = a + c$
B $3c = a + b$
C $a = b + 2c$
D $2a = b + c$

Correct answer

A. $2b = a + c$

Step-by-step solution

If the vectors are co-planar,

$\left|$ \begin{cases}a + b + 2 & a + 2b + c & - b - c \\ b + 1 & 2b & - b \\ b + 2 & 2b & 1 - b\end{cases} $\right| = 0$

Now ${R}_{1} ↔ {R}_{1} - {R}_{2}, {R}_{3} ↔ {R}_{3} - {R}_{2}$

So \left| $\begin{cases}a + 1 & a + c & - c \\ b + 1 & 2b & - b \\ 1 & 0 & 1\end{cases}$ \right| = 0 $$ ⇒ ( a + 1 ) 2b - ( a + c ) ( 2b + 1 ) - c ( - 2b ) = 0 $$ ⇒ 2ab + 2b - 2ab - a - 2bc - c + 2bc = 0 $$ ⇒ 2b - a - c = 0 ⇒ 2b = a + c$

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