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

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
\(\overrightarrow{\mathbf{a}}+\lambda \overrightarrow{\mathbf{b}}+3 \overrightarrow{\mathbf{c}}=x(-2 \overrightarrow{\mathbf{a}}+3\overrightarrow{\mathbf{b}}-4 \overrightarrow{\mathbf{c}}) \) \( + y(\overrightarrow{\mathbf{a}}-3 \overrightarrow{\mathbf{b}}+5 \overrightarrow{\mathbf{c}})\)
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, \(-1-1=\lambda\)
\(
\therefore \quad \lambda=-2
\)