If the four points, whose position vectors are 3 ^ i - 4 ^ j + 2 ^ k , ^ i + 2 ^ j - ^ k , - 2 ^ i - ^ j + 3 ^ k and 5 ^ i - 2α ^ j + 4 ^ k are coplanar, then α is equal to

Correct answer: A. (73)/(17) Let the four points be A: ( 3, - 4, 2 ) , B: ( 1, 2, - 1 ) , C: ( - 2, - 1, 3 ) and D: ( 5, - 2α, 4 ) A, B, C, D are coplanar points, then ⇒ | cases 1 - 3 & 2 + 4 & - 1 - 2 - 2 - 3 & - 1 + 4 & 3 - 2 5 - 3 & - 2α + 4 & 4 - 2 cases | = 0 ⇒ | cases - 2 & 6 & - 3 - 5 & 3 & 1 2 & - 2α + 4 & 2 cases | = 0 ⇒ - 3α + 146 = 0 ⇒ α = (73)/(17)

JEE Main 202325 JanuaryEvening Shift

Exam: JEE Main
Year: 2023
Date: 25 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2023 · 25 January · Evening Shift · Mathematics · Vector Algebra

If the four points, whose position vectors are 3 ^ i - 4 ^ j + 2 ^...

If the four points, whose position vectors are

3\overset{^}{i} - 4\overset{^}{j} + 2\overset{^}{k}, \overset{^}{i} + 2\overset{^}{j} - \overset{^}{k}, - 2\overset{^}{i} - \overset{^}{j} + 3\overset{^}{k}

and

5\overset{^}{i} - 2\alpha\overset{^}{j} + 4\overset{^}{k}

are coplanar, then

\alpha

is equal to

Options

A $\dfrac{73}{17}$
B $- \dfrac{107}{17}$
C $- \dfrac{73}{17}$
D $\dfrac{107}{17}$

Correct answer

A. $\dfrac{73}{17}$

Step-by-step solution

Let the four points be $A: ( 3, - 4, 2 )$ , $B: ( 1, 2, - 1 )$ , $C: ( - 2, - 1, 3 )$ and $D: ( 5, - 2\alpha, 4 )$

$A, B, C, D$ are coplanar points, then

$⇒ \left|$ \begin{cases}1 - 3 & 2 + 4 & - 1 - 2 \\ - 2 - 3 & - 1 + 4 & 3 - 2 \\ 5 - 3 & - 2\alpha + 4 & 4 - 2\end{cases} $\right| = 0$

$⇒ \left|$ \begin{cases}- 2 & 6 & - 3 \\ - 5 & 3 & 1 \\ 2 & - 2\alpha + 4 & 2\end{cases} $\right| = 0$

$⇒ - 3\alpha + 146 = 0$

$⇒ \alpha = \dfrac{73}{17}$

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