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If $\mathbf{a}+2 \mathbf{b}+3 \mathbf{c}=0$ and $(\mathbf{a} \times \mathbf{b})+(\mathbf{b} \times \mathbf{c})+(\mathbf{c} \times \mathbf{a})=\lambda(\mathbf{b} \times \mathbf{c})$, then the value of $\lambda$ is equal to
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Verified Answer
The correct answer is:
$6$
Given, $\mathbf{a}+2 \mathbf{b}+3 \mathbf{c}=0$
$\begin{aligned} & (a+2 b+3 c) \times c=0 \Rightarrow c \times a=2(b \times c) \\ & (a+2 b+3 c) \times b=0 \\ \Rightarrow \quad & a \times b=3(b \times c)\end{aligned}$
So,
$\begin{aligned} & (b \times c)+(c \times a)+(a \times b)=\lambda(b \times c) \\ & (b \times c)+2(b \times c)+3(b \times c)=\lambda(b \times c) \\ & 6(b+c)=\lambda(b \times c)\end{aligned}$
On comparing with coefficient of $(\mathbf{b} \times \mathbf{c})$, we get $\lambda=6$
$\begin{aligned} & (a+2 b+3 c) \times c=0 \Rightarrow c \times a=2(b \times c) \\ & (a+2 b+3 c) \times b=0 \\ \Rightarrow \quad & a \times b=3(b \times c)\end{aligned}$
So,
$\begin{aligned} & (b \times c)+(c \times a)+(a \times b)=\lambda(b \times c) \\ & (b \times c)+2(b \times c)+3(b \times c)=\lambda(b \times c) \\ & 6(b+c)=\lambda(b \times c)\end{aligned}$
On comparing with coefficient of $(\mathbf{b} \times \mathbf{c})$, we get $\lambda=6$
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