Let a =6 i + j - k and b = i + j . If c is a is vector such that | c | ≥ 6, a · c =6| c |,| c - a |=2 2 and the angle between a × b and c is 60^ , then |( a × b ) × c | is equal to:

Correct answer: C. (9)/(2)(6+ 6 ) aligned & |( a × b × c )|=| a × b || c | 3 2 & | c - a |=2 2 & |c|^2+|a|^2-2 c · a =8 & |z|^2+38-12|z|=8 & |z|^2-12|z|+30=0 & |z|= 12 ± 144-120 2 & = 12 ± 2 6 2 aligned aligned & |z|=6+ 6 & a × b =| array ccc & j & k 6 & 1 & -1 1 & 1 & 0 array | & - j +5 k & | a × b |= 27 & |( a × b) × z|= 27 (6+ 6 ) 3 2 & (9)/(2)(6+ 6 ) aligned

JEE Main 20246 AprilEvening Shift

Exam: JEE Main
Year: 2024
Date: 6 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Vector Algebra

JEE Main 2024 · 6 April · Evening Shift · Mathematics · Vector Algebra

Let a =6 i + j - k and b = i + j . If c is a is vector such that |...

Let

\overrightarrow{\mathrm{a}}=6 \hat{i}+\hat{j}-\hat{k}

and

\overrightarrow{\mathrm{b}}=\hat{i}+\hat{j}

. If

\overrightarrow{\mathrm{c}}

is a is vector such that

|\overrightarrow{\mathrm{c}}| \geq 6, \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=6|\overrightarrow{\mathrm{c}}|,|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=2 \sqrt{2}

and the angle between

\vec{a} \times \vec{b}

and

\vec{c}

60^{\circ}

, then

|(\vec{a} \times \vec{b}) \times \vec{c}|

is equal to:

Options

A $\frac{9}{2}(6-\sqrt{6})$
B $\frac{3}{2} \sqrt{6}$
C $\frac{9}{2}(6+\sqrt{6})$
D $\frac{3}{2} \sqrt{3}$

Correct answer

C. $\frac{9}{2}(6+\sqrt{6})$

Step-by-step solution

\begin{aligned} & |(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})|=|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}||\overrightarrow{\mathrm{c}}| \frac{\sqrt{3}}{2} \\ & |\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=2 \sqrt{2} \\ & |\mathrm{c}|^2+|\mathrm{a}|^2-2 \overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}=8 \\ & |\mathrm{z}|^2+38-12|\mathrm{z}|=8 \\ & |\mathrm{z}|^2-12|\mathrm{z}|+30=0 \\ & |\mathrm{z}|=\frac{12 \pm \sqrt{144-120}}{2} \\ & =\frac{12 \pm 2 \sqrt{6}}{2}\end{aligned}

\begin{aligned} & |z|=6+\sqrt{6} \\ & \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\left|\begin{array}{ccc}\hat{\ell} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 6 & 1 & -1 \\ 1 & 1 & 0\end{array}\right| \\ & \hat{\ell}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}} \\ & |\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=\sqrt{27} \\ & |(\overrightarrow{\mathrm{a}} \times \mathrm{b}) \times \mathrm{z}|=\sqrt{27}(6+\sqrt{6}) \frac{\sqrt{3}}{2} \\ & \frac{9}{2}(6+\sqrt{6})\end{aligned}

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