Let the vectors 2+a+bi^+a+2b+cj^-b+ck^, 1+bi^+2bj^-bk^ and 2+bi^+2bj^+1-bk^, ∀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, a+b+2 a+2b+c -b-c b+1 2b -b b+2 2b 1-b =0 Now R_1↔R_1-R_2, R_3↔R_3-R_2 So a+1 a+c -c b+1 2b -b 1 0 1 =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

Q35 / 90 · JEE Main 2021 (25 Jul Shift 1) · Mathematics

Let the vectors $(2 + a + b) \hat{i} + (a + 2 b + c) \hat{j} - (b + c) \hat{k}, (1 + b) \hat{i} + 2 b \hat{j} - b \hat{k}$ and $(2 + b) \hat{i} + 2 b \hat{j} + (1 - b) \hat{k}, \forall a, b, c \in R$ be co-planar. Then which of the following is true?

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

Solution

If the vectors are co-planar,

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

Now $R_{1} \leftrightarrow R_{1} - R_{2}, R_{3} \leftrightarrow R_{3} - R_{2}$

So $|\begin{matrix} a + 1 & a + c & - c \\ b + 1 & 2 b & - b \\ 1 & 0 & 1 \end{matrix}| = 0$

$\Rightarrow (a + 1) 2 b - (a + c) (2 b + 1) - c (- 2 b) = 0$

$\Rightarrow 2 a b + 2 b - 2 a b - a - 2 b c - c + 2 b c = 0$

$\Rightarrow 2 b - a - c = 0 \Rightarrow 2 b = a + c$

Hence, the correct answer is option: 'A'.

Let the vectors 2+a+bi^+a+2b+cj^-b+ck^, 1+bi^+2bj^-bk^ and 2+bi^+2bj^+1-bk^, ∀a,b,c∈R be co-planar. Then which of the following is true?

Solution

Relevant Questions

Let the vectors $(2 + a + b) \hat{i} + (a + 2 b + c) \hat{j} - (b + c) \hat{k}, (1 + b) \hat{i} + 2 b \hat{j} - b \hat{k}$ and $(2 + b) \hat{i} + 2 b \hat{j} + (1 - b) \hat{k}, \forall a, b, c \in R$ be co-planar. Then which of the following is true?