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The vectors $\mathbf{i}+2\mathbf{j}+3\mathbf{k},$ $\lambda \mathbf{i}+4\mathbf{j}+7\mathbf{k},$ $-3\mathbf{i}-2\mathbf{j}-5\mathbf{k}$ are collinear, if $\lambda$ equals
The position vectors of four points P, Q, R, S are $2\mathbf{a}+4\mathbf{c},\,$ $5\mathbf{a}+3\sqrt{3}\,\mathbf{b}+4\mathbf{c},$ $-2\sqrt{3}\mathbf{b}+\mathbf{c}$ and $2\mathbf{a}+\mathbf{c}$ respectively, then
If $\mathbf{a}=(1,\,\,-1)$ and $\mathbf{b}=(-\,2,\,m)$ are two collinear vectors, then $m =$
If three points A, B, C are collinear, whose position vectors are $\mathbf{i}-2\mathbf{j}-8\mathbf{k},\,\,5\mathbf{i}-2\mathbf{k}$ and $11\,\mathbf{i}+\,3\,\mathbf{j}+7\mathbf{k}$ respectively, then the ratio in which B divides AC is
If a and b are two non-collinear vectors and $x\,\mathbf{a}+y\,\mathbf{b}=0$
If three points A, B and C have position vectors $(1,\,x,\,3),\,\,(3,\,4,\,7)$ and $ap+bq+cr=1$ respectively and if they are collinear, then $(x,\,y)=$
$\mathbf{a}$ and $\mathbf{b}$ are two non-collinear vectors, then $x\mathbf{a}+y\mathbf{b}$ (where x and y are scalars) represents a vector which is
If $\mathbf{a}, \mathbf{b}, \mathbf{c}$, are three non-coplanar vectors such that $\mathbf{a}+\mathbf{b}+\mathbf{c}=\alpha \,\mathbf{d}$ and $\mathbf{b}+\mathbf{c}+\mathbf{d}=\beta \,\mathbf{a},$ then $\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}$ is equal to
The value of k for which the vectors $\mathbf{a}=\mathbf{i}-\mathbf{j}$ and $\mathbf{b}=-2\,\mathbf{i}+k\,\mathbf{j}$ are collinear is
$(\mathbf{a}\,.\,\mathbf{i})\,\mathbf{i}+(\mathbf{a}\,.\,\mathbf{j})\mathbf{j}+(\mathbf{a}\,.\,\mathbf{k})\,\mathbf{k}=$

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