Question

If the vectors $\overrightarrow{a}+\lambda \overrightarrow{b}+3\overrightarrow{c}$, $-2\overrightarrow{a}+3\overrightarrow{b}-4\overrightarrow{c}$ and $\overrightarrow{a}-3\overrightarrow{b}+5\overrightarrow{c}$ are coplanar, then the value of $\lambda$ is

• A. $2$
• B. $-1$
• C. $1$
• D. $-2$

Answer 1

The correct option is D $-2$

Since the given three vectors are coplanar therefore one of them should be expressible as a linear combination of the remaining two ie. there exist two scalars $x$ and $y$ such that
$\overrightarrow{a}+\lambda \overrightarrow{b}+3\overrightarrow{c}=x(2\overrightarrow{a}+3\overrightarrow{b}-4\overrightarrow{c})+y(\overrightarrow{a}-3\overrightarrow{b}+5\overrightarrow{c})$
On comparing the coefficient of $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ on both sides we get
$-2x+y=1;3x-3y=\lambda$ and $-4x+5y=3$
On solving first and third equations, we get
$x=-\frac{1}{3},y=\frac{1}{3}$
Since the vectors are coplanar, therefore these values of $x$ and $y$ also satisfy the second equation ie, $-1-1=\lambda$
$\therefore$ $\lambda =-2$