If $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}},-12 \hat{\mathbf{i}}-\hat{\mathbf{j}}-3 \hat{\mathbf{k}},-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ and $\lambda \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ are the position vectors of four coplanar points, then $\lambda=$

Let $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ be two vectors. If the orthogonal projection vector of $\mathbf{a}$ on $\mathbf{b}$ is $\mathbf{x}$ and orthogonal projection vector of $\mathbf{b}$ on $\mathbf{a}$ is $\mathbf{y}$, then $|\mathbf{x}-\mathbf{y}|=$

II. If the points with position vectors $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{k}}$, $\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ are coplanar, then the magnitude of the vector $6 \lambda \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ is

Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be three non-coplanar vectors and $L$ be the line passing through the points $\mathbf{a}-\mathbf{b}+\mathbf{c}$ and $\mathbf{b}-\mathbf{c}$. If $\pi$ is a plane passing through the points $2 \mathbf{a}-\mathbf{b}, 2 \mathbf{b}-\mathbf{c}$ and $2 c-\mathbf{a}$, then the point of intersection of $L$ and $\pi$ is