If $A(1,2,3), B(3,7,-2), C(6,7,7)$ and $D(-1,0,-1)$ are points in a plane, then the vector equation of the line passing through the centroids of $\triangle A B D$ and $\triangle A C D$ is

If $\mathbf{a}+\mathbf{b}+\mathbf{c}=0,|\mathbf{a}|=3,|\mathbf{b}|=5,|\mathbf{c}|=7$, then the angle between $\mathbf{a}$ and $\mathbf{b}$ is

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 $|\mathbf{x}-\mathbf{y}|=$