Question

Assertion :Vectors $\overrightarrow{a}=-{\lambda }^2\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-{\lambda }^2\hat{j}+\hat{k}$ and $\overrightarrow{c}=\hat{i}+\hat{j}-{\lambda }^2\hat{k}$ are coplanar for only two values of $\lambda$. Reason: If vector $\overline{x},\overline{y},\overline{z}$ are coplanar, then scalar triple product is zero.

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the correct explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

It is true that if vectors are coplanar, then their scalar triple product equal to zero, so Reason $\left(R\right)$ is true.Now scalar triple product of $\overline{a},\overline{b},\overline{c}$ is given by
$\overline{a}\cdot (\overline{b}\times \overline{c})=\left|\begin{array}{ccc}-{\lambda }^2& 1& 1\\ 1& -{\lambda }^2& 1\\ 1& 1{\lambda }^2& \end{array}\right|[\hat{i}+\hat{j}+\hat{k}]$
but $\overline{a},\overline{b},\overline{c}$ are coplaner $\therefore \overline{a}\cdot (\overline{b}\times \overline{c})=0$
$\Rightarrow {\lambda }^6-3{\lambda }^2-2=0$
$\Rightarrow ({\lambda }^2-2){({\lambda }^2+1)}^2=0$
$\Rightarrow \lambda =\pm \sqrt{2}$
So there are two values of $\lambda$ exist but no more value of $\lambda$
is possible. So Assertion $\left(A\right)$ is correct as Assertion $\left(A\right)$ &. Reason $\left(R\right)$ both are true & Reason $\left(R\right)$ is the correct explanation of Assertion $\left(A\right)$.