Question

If $a,b,c$ are non-coplanar vectors and $\lambda$ is a real number, then $\left[\lambda (\overrightarrow{a}+\overrightarrow{b}){\lambda }^2\overrightarrow{b}\lambda \overrightarrow{c}\right]=[\overrightarrow{a}\overrightarrow{b}+\overrightarrow{c}\overrightarrow{b}]$ for

• A. exactly two values of $\lambda$
• B. exactly three values of $\lambda$
• C. no value of $\lambda$
• D. exactly one value of $\lambda$

Answer 1

Answer: (C)

no value of $\lambda$

$(\lambda (\overrightarrow{a}+\overrightarrow{b})\times {\lambda }^2\overrightarrow{b}).\lambda \overrightarrow{c}={\lambda }^4((\overrightarrow{a}+\overrightarrow{b})\times \overrightarrow{b}).\overrightarrow{c}={\lambda }^4\left[abc\right]R.H.S=(\overrightarrow{a}\times (\overrightarrow{b}+\overrightarrow{c})).\overrightarrow{b}=\left[\overrightarrow{a}\overrightarrow{c}\overrightarrow{b}\right]\Rightarrow {\lambda }^4\left[abc\right]=-\left[abc\right]\Rightarrow {\lambda }^4=-1$
which is not possible.