Question

Assertion :If $\overline{a},\overline{b},\overline{c}$ are non coplanar vectors, then $2\overline{a}-\overline{b}+3\overline{c},\overline{a}+\overline{b}-2\overline{c},\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{3}c$ are also non coplanar. Reason: If vector $\overline{x},\overline{y},\overline{z}$ are non coplanar, then $\left[\overline{x}\overline{y}\overline{z}\right]\ne 0$.

• 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

Vector $\overline{a},\overline{b},\overline{c}$ are non coplaner
$\therefore \overline{a}\cdot (\overline{b}\times \overline{c})\ne 0$
Hence reason $\left(R\right)$ is correct
Now $[2\overline{a}-\overline{b}+3\overline{c}\overline{a}+\overline{b}-2\overline{c}\overline{a}+\overline{b}-3\overline{c}]$

$=\left|\begin{array}{ccc}2& -1& 3\\ 1& 1& -2\\ 1& 1& -3\end{array}\right|\left[\overline{a}\overline{b}\overline{c}\right]$

$=\left(2\right(-1)+1(-1)+3(0\left)\right)\left[\overline{a}\overline{b}\overline{c}\right]$
$=-3\left[\overline{a}\overline{b}\overline{c}\right]\ne 0$
$\Rightarrow 2\overline{a}-\overline{b}+3\overline{c},\overline{a}+\overline{b}-2\overline{c},\overline{a}+\overline{b}-3\overline{c}$ are non coplanar

So Assertion $\left(A\right)$ is correct as both Assertion $\left(A\right)$ & Reason $\left(R\right)$ are correct & Reason is the correct explanation of Assertion $\left(A\right)$.