Question

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

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

Answer 1

Answer: (C)

Let $\vec{a}=\left(a_{1}, a_{2}, a_{3}\right), \vec{b}=\left(b_{1}, b_{2}, b_{3}\right)$,
$\vec{c}=\left(c_{1}, c_{2}, c_{3}\right)$
By the given condition,
$\begin{vmatrix}\lambda\left(a_{1}+b_{1}\right)&\lambda\left(a_{2}+b_{2}\right)&\lambda\left(a_{3}+b_{3}\right)\\ \lambda^{2}b_{1}&\lambda^{2}b_{2}&\lambda^{2}b_{3}\\ \lambda c_{1}&\lambda c_{2}&\lambda c_{3}\end{vmatrix}$
$=\begin{vmatrix}a_{1}&a_{2}&a_{3}\\ b_{1}+c_{1}&b_{2}+c_{2}&b_{3}+c_{3}\\ b_{1}&b_{2}&b_{3}\end{vmatrix}$
$\Rightarrow \lambda^{4}\begin{vmatrix}a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\\ c_{1}&c_{2}&c_{3}\end{vmatrix}=\begin{vmatrix}a_{1}&a_{2}&a_{3}\\ c_{1}&c_{2}&c_{3}\\ b_{1}&b_{2}&b_{3}\end{vmatrix}$
$\Rightarrow \lambda^{4}=-1$
$\therefore $ no real value of $\lambda$ exists.