Let a→,b→,c→ are three vectors having magnitudes 1,2,3 respectively satisfy the relation |[a→ b→ c→]|=6 . If d→ is a unit vector coplanar with b→ and c→ such that b→ . d→=1 then the value of |(a→×c→) . d→|2−|(a→×c→) × d→|2

Let the angle between a→ and b→ is α and a→ × b→ and c→ is β ∴|[a→,b→,c→]|=6⇒sinαcosβ=1So⇒sinα=1,cosβ=1⇒α=900, β=00 ⇒a→,b→,c→ are mutually perpendicularagain [b→ c→ d→]2=0⇒|40109c→ . d→1c→ . d→1| =0 ⇒c→ . d→=±322 we have a→ . d→=0|a→ ×c → . d→|2= |1000933203321|=9−274=94 |(a→ ×c→) .×d→|2 =|(a→ . d→)c→−(c→ . d→)a→|2=|332a→|2=274∴ |a→×c→ . d→|2−|(a→ ×c→)×d→|2 =94−274=−184=−92

Let a→,b→,c→ are three vectors having magnitudes 1,2,3 respectively satisfy the relation |[a→ b→ c→]|=6 . If d→ is a unit vector coplanar with b→ and c→ such that b→ . d→=1 then the value of |(a→×c→) . d→|2−|(a→×c→) × d→|2

Let the angle between a→ and b→ is α and a→ × b→ and c→ is β ∴|[a→,b→,c→]|=6⇒sinαcosβ=1So⇒sinα=1,cosβ=1⇒α=900, β=00 ⇒a→,b→,c→ are mutually perpendicularagain [b→ c→ d→]2=0⇒|40109c→ . d→1c→ . d→1| =0 ⇒c→ . d→=±322 we have a→ . d→=0|a→ ×c → . d→|2= |1000933203321|=9−274=94 |(a→ ×c→) .×d→|2 =|(a→ . d→)c→−(c→ . d→)a→|2=|332a→|2=274∴ |a→×c→ . d→|2−|(a→ ×c→)×d→|2 =94−274=−184=−92

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Let $\vec{a}, \vec{b}, \vec{c}$ are three vectors having magnitudes 1,2,3 respectively satisfy the relation $| [\vec{a} \vec{b} \vec{c}] | = 6$ . If $\vec{d}$ is a unit vector coplanar with $\vec{b} a n d \vec{c}$ such that $\vec{b} . \vec{d} = 1$ then the value of ${| (\vec{a} \times \vec{c}) . \vec{d} |}^{2} - {| (\vec{a} \times \vec{c}) \times \vec{d} |}^{2}$

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$\frac{9}{2}$

$- \frac{9}{2}$

answer is C.

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Detailed Solution

Let the angle between $\vec{a} a n d \vec{b}$ $i s α a n d \vec{a} \times \vec{b} a n d \vec{c} i s β$
$\begin{array}{l} ∴ | [\vec{a}, \vec{b}, \vec{c}] | = 6 \Rightarrow \sin α \cos β = 1 \\ S o \Rightarrow \sin α = 1, \cos β = 1 \Rightarrow α = 90^{0}, β = 0^{0} \end{array}$
$\Rightarrow \vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular
$a g a i n {[\vec{b} \vec{c} \vec{d}]}^{2} = 0 \Rightarrow | \begin{matrix} 4 & 0 & 1 \\ 0 & 9 & \vec{c} . \vec{d} \\ 1 & \vec{c} . \vec{d} & 1 \end{matrix} | = 0$
$\Rightarrow \vec{c} . \vec{d} = \pm \frac{3 \sqrt{2}}{2}$ we have $\vec{a} . \vec{d} = 0$
${| \vec{a} \times \vec{c} . \vec{d} |}^{2} = | \begin{matrix} 1 & 0 & 0 \\ 0 & 9 & \frac{3 \sqrt{3}}{2} \\ 0 & \frac{3 \sqrt{3}}{2} & 1 \end{matrix} | = 9 - \frac{27}{4} = \frac{9}{4}$
${| (\vec{a} \times \vec{c}) . \times \vec{d} |}^{2}$
$= {| (\vec{a} . \vec{d}) \vec{c} - (\vec{c} . \vec{d}) \vec{a} |}^{2} = {| \frac{3 \sqrt{3}}{2} \vec{a} |}^{2} = \frac{27}{4}$
$\begin{array}{l} ∴ {| \vec{a} \times \vec{c} . \vec{d} |}^{2} - {| (\vec{a} \times \vec{c}) \times \vec{d} |}^{2} \\ = \frac{9}{4} - \frac{27}{4} = - \frac{18}{4} = - \frac{9}{2} \end{array}$