JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a} = \alpha \vec{b} - \hat{n}$ , $(\alpha \neq 0)$ and $\vec{b} \cdot \vec{c} = 12$ then $|\vec{c} \times (\vec{a} \times \vec{b})|$ is equal to:

Choose the correct answer:

Explanation

Solution

Hume expression solve karna hai: $|\vec{c} \times (\vec{a} \times \vec{b})|$

Vector Triple Product ka formula use karte hue:

\vec{c} \times (\vec{a} \times \vec{b}) = (\vec{c} \cdot \vec{b})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}

Diya gaya hai: $\vec{a} = \alpha \vec{b} - \hat{n}$

Dono taraf $\vec{c}$ se dot product lene par:

\vec{c} \cdot \vec{a} = \alpha(\vec{c} \cdot \vec{b}) - \vec{c} \cdot \hat{n}

Kyunki $\hat{n} \perp \vec{c}$ , isliye $\vec{c} \cdot \hat{n} = 0$ .

\vec{c} \cdot \vec{a} = \alpha(\vec{c} \cdot \vec{b})

Ab triple product mein value rakhte hain:

= (\vec{c} \cdot \vec{b})\vec{a} - \alpha(\vec{c} \cdot \vec{b})\vec{b}

= (\vec{c} \cdot \vec{b}) [\vec{a} - \alpha \vec{b}]

Equation $\vec{a} = \alpha \vec{b} - \hat{n}$ se hum dekh sakte hain ki $\vec{a} - \alpha \vec{b} = -\hat{n}$ .

= (\vec{c} \cdot \vec{b}) [-\hat{n}]

Magnitude lene par:

|(\vec{c} \cdot \vec{b}) (-\hat{n})| = |\vec{c} \cdot \vec{b}| \times |\hat{n}|

Kyunki $\vec{b} \cdot \vec{c} = 12$ aur $|\hat{n}| = 1$ :

12 \times 1 = 12

Correct Option: (4)

Choose the correct answer:

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