Let $a = i + 2j + k$,$b = i - j + k$,$c = i + j - k$. $A$ vector in the plane of $a$ and $b$ has projection $\frac{1}{\sqrt{3}}$ on $c$. Then,one such vector is

If $a, b$ and $c$ are three non-zero vectors,no two of which are collinear. If the vector $a + 2b$ is collinear with $c$ and $b + 3c$ is collinear with $a$,then ($\lambda$ being some non-zero scalar) $a + 2b + 6c$ is equal to

Let $\bar{u}, \bar{v}, \bar{w}$ be vectors such that $|\bar{u}|=1, |\bar{v}|=2, |\bar{w}|=3$. If the projection of $\bar{v}$ on $\bar{u}$ is equal to that of $\bar{w}$ on $\bar{u}$,and the vectors $\bar{v}, \bar{w}$ are perpendicular to each other,then $|\bar{u}-\bar{v}+\bar{w}|=$

Find the angle between the lines whose direction ratios are $a, b, c$ and $b-c, c-a, a-b$. (in $^{\circ}$)

If $|a|=3, |b|=4$ and the angle between $a$ and $b$ is $120^{\circ}$,then $|4a+3b|$ is equal to

If $a = i + j$ and $b = 2i - k$ are two vectors, then the point of intersection of two lines $r \times a = b \times a$ and $r \times b = a \times b$ is