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 should exist two scalars x and y such that
$\overrightarrow{a}+\lambda \overrightarrow{b}+3\overrightarrow{c}=x(-2\overrightarrow{a}+3\overrightarrow{b}-4\overrightarrow{c})+y(\overrightarrow{a}-3\overrightarrow{b}+5\overrightarrow{c})$
$\Rightarrow -2x+y=1;3x-3y=\lambda$ ; and 3 =-4x +5y
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 third equation
$\lambda =-2$