If $35 \hat{\mathbf{i}}+14 \hat{\mathbf{j}}-77 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+5…

If $35 \hat{\mathbf{i}}+14 \hat{\mathbf{j}}-77 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ and $5 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ are coplanar, then $\lambda$ is equal to
  1. 11
  2. -11
  3. -10
  4. 10

Solution

Given, $35 \hat{\mathbf{i}}+14 \hat{\mathbf{j}}-77 \hat{\mathbf{k}}$, $2 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ and $5 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ are coplanar. Then, $\quad\left|\begin{array}{ccc}35 & 14 & -77 \\ 2 & 7 & 5 \\ 5 & 2 & \lambda\end{array}\right|=0$ $ \begin{aligned} & 35(7 \lambda-10)-14(2 \lambda-25)-77(4-35) & =0 \\ \Rightarrow & 203 \lambda+2233 & =0 \\ \Rightarrow & 203(\lambda+11) & =0 \\ \Rightarrow & \lambda & =-11 \end{aligned} $

Asked in: AP EAMCET 2021 (23 Aug Shift 1)

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