Let → p = 2 ^ i + 3 ^ j + ^ k and → q = ^ i + 2 ^ j + ^ k be two vectors. If a vector → r = (α ^ i + β ^ j + γ ^ k ) is perpendicular to each of the vectors ( → p + → q ) and ( → p - → q ) , and | → r | = 3 , then | α |

Correct answer: 0 Given, → p = 2 ^ i + 3 ^ j + ^ k → q = ^ i + 2 ^ j + ^ k So, → p + → q = 3 ^ i + 5 ^ j + 2 ^ k and → p - → q = ^ i + ^ j Now, ( → p + → q ) × ( → p - → q ) = | cases ^ i & ^ j & ^ k 3 & 5 & 2 1 & 1 & 0 cases | = ^ i (0 - 2) - ^ j (0 - 2) + ^ k (3 - 5) = - 2 ^ i + 2 ^ j - 2 ^ k ⇒ → r = ± 3 ( ( → p + → q ) × ( → p - → q ) ) | ( → p + → q ) × ( → p - → q ) | = ± 3 ( - 2 ^ i + 2 ^ j - 2 ^ k ) 2 ^ 2 + 2 ^ 2 + 2 ^ 2 → r = ± (- ^ i + ^ j - ^ k ) According to question → r = α ^ i + β ^ j + γ ^ k So | α | = 1, | β | = 1, | γ | = 1 ⇒ | α | + | β | + | γ | = 3

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 → p = 2 ^ i + 3 ^ j + ^ k and → q = ^ i + 2 ^ j + ^ k be two...

Let $\overset{\to}{p} = 2\overset{^}{i} + 3\overset{^}{j} + \overset{^}{k} and$ \overset{\to}{q} = \overset{^}{i} + 2\overset{^}{j} + \overset{^}{k} $be two vectors. If a vector$ \overset{\to}{r} = $\left(\alpha\overset{^}{i} + \beta\overset{^}{j} + \gamma\overset{^}{k}\right)$ is perpendicular to each of the vectors $( \overset{\to}{p} + \overset{\to}{q} )$ and $( \overset{\to}{p} - \overset{\to}{q} )$ , and $| \overset{\to}{r} | = \sqrt{3}$ , then $| \alpha | + | \beta | + | \gamma |$ is equal to

Correct answer

Step-by-step solution

Given,

$\overset{\to}{p} = 2\overset{^}{i} + 3\overset{^}{j} + \overset{^}{k}$

$\overset{\to}{q} = \overset{^}{i} + 2\overset{^}{j} + \overset{^}{k}$

So, $\overset{\to}{p} + \overset{\to}{q} = 3\overset{^}{i} + 5\overset{^}{j} + 2\overset{^}{k} \text{ and } \overset{\to}{p} - \overset{\to}{q} = \overset{^}{i} + \overset{^}{j}$

Now, \left(\overset{\to}{p} + \overset{\to}{q}\right) \times \left(\overset{\to}{p} - \overset{\to}{q}\right) = \left| $\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ 3 & 5 & 2 \\ 1 & 1 & 0\end{cases}$ \right| $$ = \overset{^}{i}\left(0 - 2\right) - \overset{^}{j}\left(0 - 2\right) + \overset{^}{k}\left(3 - 5\right) $$ = - 2\overset{^}{i} + 2\overset{^}{j} - 2\overset{^}{k} $$ ⇒ \overset{\to}{r} = \pm \sqrt{3}\dfrac{( ( \overset{\to}{p} + \overset{\to}{q} ) \times ( \overset{\to}{p} - \overset{\to}{q} ) )}{| ( \overset{\to}{p} + \overset{\to}{q} ) \times ( \overset{\to}{p} - \overset{\to}{q} ) |} = \pm \dfrac{\sqrt{3} ( - 2\overset{^}{i} + 2\overset{^}{j} - 2\overset{^}{k} )}{\sqrt{{2}^{2} + {2}^{2} + {2}^{2}}} $$ \overset{\to}{r} = \pm \left(- \overset{^}{i} + \overset{^}{j} - \overset{^}{k}\right) $According to question $ \overset{\to}{r} = \alpha\overset{^}{i} + \beta\overset{^}{j} + \gamma\overset{^}{k} $So$ | \alpha | = 1, | \beta | = 1, | \gamma | = 1 $$ ⇒ | \alpha | + | \beta | + | \gamma | = 3$

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