If vector a→,b→,c→ are linearly independent and (d→×b→)×(c→×a→)+(d→×c→)×(a→×b→)+(d→×a→)×(b→×c→)=r→

r → is collinear with d → , d → . ( a → × r → ) = 0 , r → is a linear combination of a → , b → and c →

If vector a→,b→,c→ are linearly independent and (d→×b→)×(c→×a→)+(d→×c→)×(a→×b→)+(d→×a→)×(b→×c→)=r→

Since a → , b → , c → are linearly independent ⇒ [ a → b → c → ] ≠ 0 ⇒ d → can be written as linear combination of a → , b → and c → d → = [ d → b → c → ] a → + [ d → c → a → ] b → + [ d → a → b → ] c → [ a → b → c → ] Now, r → = [ d → b → a → ] c → − [ d → b → c → ] a → + [ d → c → b → ] a → − [ d → c → a → ] b → + [ d → a → c → ] b → + [ d → a → c → ] b → − [ d → a → b → ] c → = 2 { [ d → c → b → ] a → + [ d → a → c → ] b → + [ d → b → a → ] c → } = − 2 [ a → b → c → ] d → ⇒ r → = − 2 [ a → b → c → ] d → ( ( b → + c → ) × ( b → + d → ) ) × ( c → × d → ) ⋅ c → = ( ( c → × b → + c → × d → + b → × d → ) × ( c → × d → ) ) ⋅ c → = ( [ c → b → d → ] c → − [ b → d → c → ] d → ) ⋅ c → = [ b → d → c → ] | d → | 2

d → = 2 [ a → b → c → ] r →

r → is a linear combination of a → , b → and c →

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If vector $\vec{a}, \vec{b}, \vec{c}$ are linearly independent and $(\vec{d} \times \vec{b}) \times (\vec{c} \times \vec{a}) + (\vec{d} \times \vec{c}) \times (\vec{a} \times \vec{b}) + (\vec{d} \times \vec{a}) \times (\vec{b} \times \vec{c}) = \vec{r}$

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$\vec{r}$ is collinear with $\vec{d}$

$\vec{d} . (\vec{a} \times \vec{r}) = 0$

$\vec{d} = 2 [\vec{a} \vec{b} \vec{c}] \vec{r}$

$\vec{r}$ is a linear combination of $\vec{a}, \vec{b}$ and $\vec{c}$

answer is A, B, D.

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Detailed Solution

Since $\vec{a}, \vec{b}, \vec{c}$ are linearly independent $\Rightarrow [\vec{a} \vec{b} \vec{c}] \neq 0$
$\Rightarrow \vec{d}$ can be written as linear combination of $\vec{a}, \vec{b}$ and $\vec{c}$
$\vec{d} = \frac{[\vec{d} \vec{b} \vec{c}] \vec{a} + [\vec{d} \vec{c} \vec{a}] \vec{b} + [\vec{d} \vec{a} \vec{b}] \vec{c}}{[\vec{a} \vec{b} \vec{c}]}$
Now, $\vec{r} = [\vec{d} \vec{b} \vec{a}] \vec{c} - [\vec{d} \vec{b} \vec{c}] \vec{a} + [\vec{d} \vec{c} \vec{b}] \vec{a} - [\vec{d} \vec{c} \vec{a}] \vec{b} + [\vec{d} \vec{a} \vec{c}] \vec{b} + [\vec{d} \vec{a} \vec{c}] \vec{b} - [\vec{d} \vec{a} \vec{b}] \vec{c}$
$= 2 {[\vec{d} \vec{c} \vec{b}] \vec{a} + [\vec{d} \vec{a} \vec{c}] \vec{b} + [\vec{d} \vec{b} \vec{a}] \vec{c}} = - 2 [\vec{a} \vec{b} \vec{c}] \vec{d} \Rightarrow \vec{r} = - 2 [\vec{a} \vec{b} \vec{c}] \vec{d} ((\vec{b} + \vec{c}) \times (\vec{b} + \vec{d})) \times (\vec{c} \times \vec{d}) \cdot \vec{c} = ((\vec{c} \times \vec{b} + \vec{c} \times \vec{d} + \vec{b} \times \vec{d}) \times (\vec{c} \times \vec{d})) \cdot \vec{c} = ([\vec{c} \vec{b} \vec{d}] \vec{c} - [\vec{b} \vec{d} \vec{c}] \vec{d}) \cdot \vec{c} = [\vec{b} \vec{d} \vec{c}] {| \vec{d} |}^{2}$

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