QJEEM21Mada48: JEE Main 2021 Physics - Class 11 JEEM21Mada48 Solution: Let and be two vectors. If a vector is perpendicular to each of …

Question

Verified Solution • 2025–26 Session moderate

Let

\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k

and

\overrightarrow q = \widehat i + 2\widehat j + \widehat k

be two vectors. If a vector

\overrightarrow r = (\alpha \widehat i + \beta \widehat j + \gamma \widehat k)

is perpendicular to each of the vectors (

(\overrightarrow p + \overrightarrow q )

and

(\overrightarrow p - \overrightarrow q )

, and

\left| {\overrightarrow r } \right| = \sqrt 3

, then

\left| \alpha \right| + \left| \beta \right| + \left| \gamma \right|

is equal to _______________.

Accepted Answer

\overrightarrow p = 2\widehat i + 3\widehat j + \widehat k

(Given )

\overrightarrow q = \widehat i + 2\widehat j + \widehat k

Now,

(\vec{p} + \vec{q}) \times (\vec{p} - \vec{q}) = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 5 & 2 \\ 1 & 1 & 0 \end{matrix} |

= - 2\widehat i - 2\widehat j - 2\widehat k

\Rightarrow \overrightarrow r = \pm \sqrt 3 {{\left( {(\overrightarrow p + \overrightarrow q ) \times (\overrightarrow p - \overrightarrow q )} \right)} \over {\left| {(\overrightarrow p + \overrightarrow q ) \times (\overrightarrow p - \overrightarrow q )} \right|}} = \pm {{\sqrt 3 \left( { - 2\widehat i - 2\widehat j - 2\widehat k} \right)} \over {\sqrt {{2^2} + {2^2} + {2^2}} }}

\overrightarrow r = \pm \left( { - \widehat i - \widehat j - \widehat k} \right)

According to question

\overrightarrow r = \alpha \widehat i + \beta \widehat j + \gamma \widehat k

So, |

\alpha

| = 1, |

\beta

| = 1, |

\gamma

| = 1

\Rightarrow

\left| \alpha \right| + \left| \beta \right| + \left| \gamma \right|

= 3

Step-by-Step Verified Solution