If the planes → r · (2 ^ i - λ ^ j + 3 ^ k ) = 0 and → r · (λ ^ i + 5 ^ j - ^ k ) = 5 are perpendicular to each other, then the value of λ ^ 2 + λ is

Correct answer: A. 0 We have, → r · (λ ^ i + 5 ^ j - ^ k ) = 5 → r · (2 ^ i - λ ^ j + 3 ^ k ) = 0 We know that two planes~ → r · → n 1 = d _ 1 ~and& → r · → n 2 = d _ 2 ~are perpendicular if → n 1 · → n 2 = 0 Therefore, (λ ^ i + 5 ^ j - ^ k ) · (2 ^ i - λ ^ j + 3 ^ k ) = 0 ⇒ 2λ - 5λ - 3 = 0 ⇒ - 3λ - 3 = 0 ⇒ λ = - 1 Therefore,~ λ ^ 2 + λ = 0

BITSAT 2018

Exam: BITSAT
Year: 2018
Subject: Mathematics
Chapter: Three Dimensional Geometry

BITSAT 2018 · Mathematics · Three Dimensional Geometry

If the planes → r · (2 ^ i - λ ^ j + 3 ^ k ) = 0 and → r · (λ ^ i +...

If the planes

\overset{→}{r} \cdot \left(2\overset{^}{i} - \lambda\overset{^}{j} + 3\overset{^}{k}\right) = 0

and

\overset{→}{r} \cdot \left(\lambda\overset{^}{i} + 5\overset{^}{j} - \overset{^}{k}\right) = 5

are perpendicular to each other, then the value of

{λ}^{2} + \lambda

Options

A $0$
B $2$
C $1$
D $3$

Correct answer

A. $0$

Step-by-step solution

We have,

$\overset{→}{r} \cdot \left(\lambda\overset{^}{i} + 5\overset{^}{j} - \overset{^}{k}\right) = 5$

$\overset{→}{r} \cdot \left(2\overset{^}{i} - \lambda\overset{^}{j} + 3\overset{^}{k}\right) = 0$

We know that two planes~ $\overset{→}{r} \cdot \overset{→}{n}1 = {d}_{1}$ ~and& $\overset{→}{r} \cdot \overset{→}{n}2 = {d}_{2}$ ~are perpendicular if

$\overset{→}{n}1 \cdot \overset{→}{n}2 = 0$

Therefore,

\left(\lambda\overset{^}{i} + 5\overset{^}{j} - \overset{^}{k}\right) \cdot \left(2\overset{^}{i} - \lambda\overset{^}{j} + 3\overset{^}{k}\right) = 0 $$ ⇒ 2\lambda - 5\lambda - 3 = 0 $$ ⇒ - 3\lambda - 3 = 0 $$ ⇒ \lambda = - 1 $Therefore,~$ {λ}^{2} + \lambda = 0

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