Q64 / 150 · MHT CET 2007 · Mathematics
If the vectors $\overrightarrow{\mathbf{a}}+\lambda \overrightarrow{\mathbf{b}}+3 \overrightarrow{\mathbf{c}},-2 \overrightarrow{\mathbf{a}}+3 \overrightarrow{\mathbf{b}}-4 \overrightarrow{\mathbf{c}}$
and $\overrightarrow{\mathbf{a}}-3 \overrightarrow{\mathbf{b}}+5 \overrightarrow{\mathbf{c}}$ are coplanar, then the value of $\lambda$ is
- A2
- B$-1$
- C1
- Correct answerD$-2$
Solution
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
$$
\begin{aligned}
\overrightarrow{\mathbf{a}}+\lambda \overrightarrow{\mathbf{b}}+3 \overrightarrow{\mathbf{c}}=x(-2 \overrightarrow{\mathbf{a}}+3&\overrightarrow{\mathbf{b}}-4 \overrightarrow{\mathbf{c}}) \\
+& y(\overrightarrow{\mathbf{a}}-3 \overrightarrow{\mathbf{b}}+5 \overrightarrow{\mathbf{c}})
\end{aligned}
$$
On comparing the coefficient of $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ on both sides, we get
$$
-2 x+y=1 ; 3 x-3 y=\lambda
$$
and $-4 x+5 y=3$
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, $-1-1=\lambda$
$$
\therefore \quad \lambda=-2
$$
Hence, the correct answer is option: 'D'.