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Question: If \(\mathbf{p} = \frac{\mathbf{b} \times \mathbf{c}}{\lbrack\mathbf{abc}\rbrack},\mathbf{q} = \frac...

If p=b×c[abc],q=c×a[abc],r=a×b[abc],\mathbf{p} = \frac{\mathbf{b} \times \mathbf{c}}{\lbrack\mathbf{abc}\rbrack},\mathbf{q} = \frac{\mathbf{c} \times \mathbf{a}}{\lbrack\mathbf{abc}\rbrack},\mathbf{r} = \frac{\mathbf{a} \times \mathbf{b}}{\lbrack\mathbf{abc}\rbrack}, where a, b, c are three non-coplanar vectors, then the value of (a+b+c).(p+q+r)(\mathbf{a} + \mathbf{b} + \mathbf{c}).(\mathbf{p} + \mathbf{q} + \mathbf{r}) is given by

A

3

B

2

C

1

D

0

Answer

3

Explanation

Solution

p+q+r=b×c+c×a+a×b[abc]\mathbf{p} + \mathbf{q} + \mathbf{r} = \frac{\mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} + \mathbf{a} \times \mathbf{b}}{\lbrack\mathbf{abc}\rbrack}

(a+b+c).(p+q+r)=[abc]+[bca]+[cab][abc]=3(\mathbf{a} + \mathbf{b} + \mathbf{c}).(\mathbf{p} + \mathbf{q} + \mathbf{r}) = \frac{\lbrack\mathbf{abc}\rbrack + \lbrack\mathbf{bca}\rbrack + \lbrack\mathbf{cab}\rbrack}{\lbrack\mathbf{abc}\rbrack} = 3.