Question

The value of $x\in R$ for which vectors $\overrightarrow{a}=(1,-2,1),\overrightarrow{b}=(-2,3,-4),\overrightarrow{c}=(1,-1,x)$ form a linearly dependent system, is equal to

Answer 1

As, $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ form a linearly dependent system. So one vector can be expressed as linear combination of other two.
$\Rightarrow \overrightarrow{a}={\lambda }_1\overrightarrow{b}+{\lambda }_2\overrightarrow{c}$
(where ${\lambda }_1,{\lambda }_2$ are scalars.)
$\Rightarrow \hat{i}-2\hat{j}+\hat{k}={\lambda }_1(-2\hat{i}+3\hat{j}-4\hat{k})+{\lambda }_2(\hat{i}-\hat{j}+x\hat{k})$
equating cofficients of $\hat{i},\hat{j}$ and $\hat{k}$ both sides,
$1=-2{\lambda }_1+{\lambda }_2\cdots \left(i\right)-2=3{\lambda }_1-{\lambda }_2\cdots \left(ii\right)1=-4{\lambda }_1+x{\lambda }_2\cdots \left(iii\right)$
on solving $\left(i\right)$ and $\left(ii\right):$ we get ${\lambda }_1=-1,{\lambda }_2=-1$
From $\left(iii\right),$ we get
$x=3$