If p→= b→× c→a→ b→ c→ , q→ = c→ × a→a→ b→ c→ and r→ = a→× b→ a→ b→ c→, where a→,b→ and c→ are three non-coplanar vectors then the value of the expression a→ + b→ + c→.p→ + q→ + r→ is___

We have a→. p→= a→.b→× c→a→ b→ c→ = a→.b→× c→a→ b→ c→ = 1 a→. q→ = a→. c→ × a→a→ b→ c→ = a→c→ a→a→ b→ c→ = 0Similarly a→ . r→ = 0, b→.p→ = 0, b→.q→ = 1, b→.r→ = 0 c→.p→ = 0, c→.q→ = 0 and c→.r→ = 1 =a→ .p→+ a→ .q→ + a→.r→+ b→.p→ + b→ .q→ + b→.r→ + c→ .p→ + c→.q→ + c→.r→ = 1 + 1 + 1 = 3

If p→= b→× c→a→ b→ c→ , q→ = c→ × a→a→ b→ c→ and r→ = a→× b→ a→ b→ c→, where a→,b→ and c→ are three non-coplanar vectors then the value of the expression a→ + b→ + c→.p→ + q→ + r→ is___

We have a→. p→= a→.b→× c→a→ b→ c→ = a→.b→× c→a→ b→ c→ = 1 a→. q→ = a→. c→ × a→a→ b→ c→ = a→c→ a→a→ b→ c→ = 0Similarly a→ . r→ = 0, b→.p→ = 0, b→.q→ = 1, b→.r→ = 0 c→.p→ = 0, c→.q→ = 0 and c→.r→ = 1 =a→ .p→+ a→ .q→ + a→.r→+ b→.p→ + b→ .q→ + b→.r→ + c→ .p→ + c→.q→ + c→.r→ = 1 + 1 + 1 = 3

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Offline Centres

If $\vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]} a n d \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]}$ , where $\vec{a}, \vec{b} a n d \vec{c}$ are three non-coplanar vectors then the value of the expression $(\vec{a} + \vec{b} + \vec{c}) . (\vec{p} + \vec{q} + \vec{r})$ is___

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Detailed Solution

We have $\vec{a} . \vec{p} = \vec{a} . \frac{(\vec{b} \times \vec{c})}{[\vec{a} \vec{b} \vec{c}]} = \frac{\vec{a} . (\vec{b} \times \vec{c})}{[\vec{a} \vec{b} \vec{c}]} = 1 \vec{a} . \vec{q} = \vec{a} . \frac{(\vec{c} \times \vec{a})}{[\vec{a} \vec{b} \vec{c}]} = \frac{[\vec{a} \vec{c} \vec{a}]}{[\vec{a} \vec{b} \vec{c}]} = 0$

Similarly $\vec{a} . \vec{r} = 0, \vec{b} . \vec{p} = 0, \vec{b} . \vec{q} = 1, \vec{b} . \vec{r} = 0 \vec{c} . \vec{p} = 0, \vec{c} . \vec{q} = 0 and \vec{c} . \vec{r} = 1 = \vec{a} . \vec{p} + \vec{a} . \vec{q} + \vec{a} . \vec{r} + \vec{b} . \vec{p} + \vec{b} . \vec{q} + \vec{b} . \vec{r} + \vec{c} . \vec{p} + \vec{c} . \vec{q} + \vec{c} . \vec{r} = 1 + 1 + 1 = 3$