If $3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\sqrt{3} \hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{k}}, \sqrt{3} \hat{\mathbf{i}}+\sqrt{3} \hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ are coplanar, then $\lambda$ is equal to

Question

Aicharya · Accepted Answer

Let $\mathbf{a}=3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\sqrt{3} \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{k}}$ and $ \mathbf{c}=\sqrt{3} \hat{\mathbf{i}}+\sqrt{3} \hat{\mathbf{j}}+\lambda \hat{\mathbf{k}} $ Since, these vectors are coplanar $ \begin{aligned} & 	herefore \quad \mathbf{a} \cdot(\mathbf{b} 	imes \mathbf{c})=0 \ & \Rightarrow \quad\left|\begin{array}{ccc} 3 & 3 & \sqrt{3} \ 1 & 0 & 1 \ \sqrt{3} & \sqrt{3} & \lambda \end{array}ight|=0 \ & \Rightarrow \quad 3(0-\sqrt{3})-3(\lambda-\sqrt{3})+\sqrt{3}(\sqrt{3}-0)=0 \ & \Rightarrow \quad-3 \sqrt{3}-3 \lambda+3 \sqrt{3}+3=0 \ & \Rightarrow \quad \lambda=1 \ & \end{aligned} $

If $3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\sqrt{3} \hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{k}},…

Solution