Q84 / 180 · KCET 2020 · Mathematics

If the vectors $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are coplanar, then the value of $\lambda$ is

  • Correct answer
    A
    6
  • B
    $-5$
  • C
    -6
  • D
    5

Solution

Given vectors $2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ are coplanar $\left|\begin{array}{ccc}2 & -3 & 4 \\ 2 & 1 & -1 \\ \lambda & -1 & 2\end{array}\right|=0$ $\Rightarrow \quad 2(2-1)+3(4+\lambda)+4(-2-\lambda)=0$ $\Rightarrow \quad 2+12+3 \lambda-8-4 \lambda=0$ $\Rightarrow \quad 6-\lambda=0$ $\Rightarrow \quad \lambda=6$

Hence, the correct answer is option: 'A'.

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