If $\vec{a}=\hat{i}+2 \hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+4 \hat{k}$ and $\vec{c}=\hat{i}+\hat{j}+\hat{k}$ are such that $\vec{a}+\lambda \vec{b}$ is perpendicular to $\vec{c}$, then the value of $\lambda$ is

Consider the following statements :

Statement (I) : If either $|\vec{a}|=0$ or $|\vec{b}|=0$, then $\vec{a} \cdot \vec{b}=0$

Statement (II) : If $\vec{a} \times \vec{b}=\overrightarrow{0}$, then a is perpendicular to $b$. Which of the following is correct?

The vectors $\mathbf{A B}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{k}}$ and $\mathbf{A C}=5 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ are the sides of a $\triangle A B C$, The length of the median through $A$ is