The magnitude of the projection of the vector $\vec{a} = -\hat{i} + 2\hat{j} - \hat{k}$ on the unit vector $\hat{i}$ is . . . . . . .

The sum of the three vectors determined by the medians of a triangle directed from the vertices is

$ABCD$ is a parallelogram such that $L$ is the mid-point of $BC$. Then,$\vec{AL}$ is equal to:

$R$ 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

$PQRS$ is a quadrilateral and $PQ=a, QR=b, SP=a-b$. $M$ is the mid-point of $QR$ and $X$ is a point on $SM$ such that $SX=\frac{4}{5}SM$. If $SM=m(4a-b)$ and $SX=n(4a-b)$,then $m+n=$

$A, B, C, D$ are any four points. If $E$ and $F$ are midpoints of $AC$ and $BD$ respectively, then $\vec{AB} + \vec{CB} + \vec{CD} + \vec{AD} =$