If $a = i - 2j$ and $b = 2i + \lambda j$ are parallel,then $\lambda$ is

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $|\vec{a}|=3, |\vec{b}|=5, |\vec{c}|=7$,then $|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2$ does not exceed:

If $a = 2i + j - 8k$ and $b = i + 3j - 4k,$ then the magnitude of $a + b = $

If $A(1, 2, 1), B(2, 3, 2), C(2, 1, 3), D(3, 2, 4)$,then the directions of $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are $.......$

If $\overline{r} = -4 \hat{i} - 6 \hat{j} - 2 \hat{k}$ is a linear combination of the vectors $\overline{a} = -\hat{i} - 4 \hat{j} + 3 \hat{k}$ and $\overline{b} = -8 \hat{i} - \hat{j} + 3 \hat{k}$,then which of the following is true?

$R$ divides the line joining two points $P$ and $Q$ whose position vectors are $\hat{i}+2 \hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively in the ratio $2: 1$ externally. $S$ divides $PQ$ internally in the ratio $2: 1$. Then,the position vector of the midpoint of the line joining $R$ and $S$ is