KCET · Maths · Vector Algebra
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
- A \(3\)
- B \(4\)
- C \(6\)
- D \(2\)
Answer & Solution
Correct Answer
(C) \(6\)
Step-by-step Solution
Detailed explanation
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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