Let a→,b→ and c→ be three non-coplanar vectors and p→,q→ and r→ the vectors defined by the relations p→=b→×c→[a→b→c→],q→=c→×a→[a→b→c→] and r→=a→×b→[a→b→c→] Then the value of the expression (a→+b)⋅p→+(b→+c→)⋅q→+(c→+a→)⋅r→ is

Given that a→,b→,c→ are non-coplanar. Therefore, [a→b→c→]≠0ALSO ,p→=b→×c→[a→b→c→], q→=c→×a→[a→b→c→],r→=a→×b→[a→b→c→]now,(a→+b→)⋅p→+(b→+c→)⋅q→+(c→+a→)⋅r→= (a→+b→)⋅b→×c→[a→b→c→]+(b→+c→)⋅c→×a→[a→b→c→]+(c→+a→)⋅a→×b→[a→b→c→]=a→⋅b→×c→[a→b→c→]+b→⋅c→×a→[a→b→c→]+c→⋅a→×b→[a→b→c→][∵b→⋅b→×c→=c→⋅c→×a→=a→⋅a→×b→=0]=[a→b→c→][a→b→c→]+[a→b→c→][a→b→c→]+[a→b→c→][a→b→c→]=1+1+1=3

Let a→,b→ and c→ be three non-coplanar vectors and p→,q→ and r→ the vectors defined by the relations p→=b→×c→[a→b→c→],q→=c→×a→[a→b→c→] and r→=a→×b→[a→b→c→] Then the value of the expression (a→+b)⋅p→+(b→+c→)⋅q→+(c→+a→)⋅r→ is

Given that a→,b→,c→ are non-coplanar. Therefore, [a→b→c→]≠0ALSO ,p→=b→×c→[a→b→c→], q→=c→×a→[a→b→c→],r→=a→×b→[a→b→c→]now,(a→+b→)⋅p→+(b→+c→)⋅q→+(c→+a→)⋅r→= (a→+b→)⋅b→×c→[a→b→c→]+(b→+c→)⋅c→×a→[a→b→c→]+(c→+a→)⋅a→×b→[a→b→c→]=a→⋅b→×c→[a→b→c→]+b→⋅c→×a→[a→b→c→]+c→⋅a→×b→[a→b→c→][∵b→⋅b→×c→=c→⋅c→×a→=a→⋅a→×b→=0]=[a→b→c→][a→b→c→]+[a→b→c→][a→b→c→]+[a→b→c→][a→b→c→]=1+1+1=3

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$Let \vec{a}, \vec{b} and \vec{c} be three non-coplanar vectors and \vec{p}, \vec{q} and \vec{r} the vectors defined by the relations$
$\vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]} and \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]} Then the value of the expression (\vec{a} + b) \cdot \vec{p} + (\vec{b} + \vec{c}) \cdot \vec{q} + (\vec{c} + \vec{a}) \cdot \vec{r} is$

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Detailed Solution

$Given that \vec{a}, \vec{b}, \vec{c} are non-coplanar. Therefore, [\vec{a} \vec{b} \vec{c}] \neq 0$

$\begin{array}{l} A L S O, \vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]}, \\ \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]} \end{array}$

$\begin{array}{l} now, (\vec{a} + \vec{b}) \cdot \vec{p} + (\vec{b} + \vec{c}) \cdot \vec{q} + (\vec{c} + \vec{a}) \cdot \vec{r} \\ = (\vec{a} + \vec{b}) \cdot \frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]} + (\vec{b} + \vec{c}) \cdot \frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]} + (\vec{c} + \vec{a}) \cdot \frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]} \end{array}$

$= \frac{\vec{a} \cdot \vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]} + \frac{\vec{b} \cdot \vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]} + \frac{\vec{c} \cdot \vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]}$

$\begin{array}{l} [∵ \vec{b} \cdot \vec{b} \times \vec{c} = \vec{c} \cdot \vec{c} \times \vec{a} = \vec{a} \cdot \vec{a} \times \vec{b} = 0] \\ = \frac{[\vec{a} \vec{b} \vec{c}]}{[\vec{a} \vec{b} \vec{c}]} + \frac{[\vec{a} \vec{b} \vec{c}]}{[\vec{a} \vec{b} \vec{c}]} + \frac{[\vec{a} \vec{b} \vec{c}]}{[\vec{a} \vec{b} \vec{c}]} \\ = 1 + 1 + 1 \\ = 3 \end{array}$