Let a, b and c be distinct positive numbers. If the vectors a ^ i + a ^ j + c ^ k , ^ i + ^ k and c ^ i + c ^ j + b ^ k are co-planar, then c is equal to:

Correct answer: D. ab Given vectors a ^ i + a ^ j + c ^ k , ^ i + ^ k and c ^ i + c ^ j + b ^ k are co-planar, then | cases a & a & c 1 & 0 & 1 c & c & b cases | = 0 ⇒ a(0 - c) - a(b - c) + c(c - 0) = 0 ⇒ c ^ 2 = ab ⇒ c = ab .

JEE Main 202125 JulyEvening Shift

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

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

Let a, b and c be distinct positive numbers. If the vectors a ^ i +...

Let

a, b

and

c

be distinct positive numbers. If the vectors

a\overset{^}{i} + a\overset{^}{j} + c\overset{^}{k}, \overset{^}{i} + \overset{^}{k}

and

c\overset{^}{i} + c\overset{^}{j} + b\overset{^}{k}

are co-planar, then

c

is equal to:

Options

A $\dfrac{2}{\dfrac{1}{a} + \dfrac{1}{b}}$
B $\dfrac{a + b}{2}$
C $\dfrac{1}{a} + \dfrac{1}{b}$
D $\sqrt{ab}$

Correct answer

D. $\sqrt{ab}$

Step-by-step solution

Given vectors $a\overset{^}{i} + a\overset{^}{j} + c\overset{^}{k}, \overset{^}{i} + \overset{^}{k} and$ c\overset{^}{i} + c\overset{^}{j} + b\overset{^}{k} $are co-planar, then</p><p>\left|$ \begin{cases}a & a & c \\ 1 & 0 & 1 \\ c & c & b\end{cases} $\right| = 0$

$⇒ a\left(0 - c\right) - a\left(b - c\right) + c\left(c - 0\right) = 0$

$⇒ {c}^{2} = ab$

$⇒ c = \sqrt{ab} .</p>$

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