The position vector of a point at a distance of $3\sqrt{11}$ units from $i - j + 2k$ on a line passing through the points $i - j + 2k$ and $3i + j + k$ is

Let $a$ and $b$ be unit vectors with $\theta$ as the acute angle between them. If $\frac{1}{2}|a-b|=\sin(\lambda \theta)$,then $4 \lambda^2=$

The points with position vectors $10\,i + 3\,j$,$12\,i - 5\,j$,and $a\,i + 11\,j$ are collinear,if $a = $

Find the vector joining the points $P(2, 3, 0)$ and $Q(-1, -2, -4)$ directed from $P$ to $Q$.

Let $\vec{i}+\vec{j}+\vec{k}$, $a_1 \vec{i}+b_1 \vec{j}+c_1 \vec{k}$, $a_2 \vec{i}+b_2 \vec{j}+c_2 \vec{k}$, and $a_3 \vec{i}+b_3 \vec{j}+c_3 \vec{k}$ be the position vectors of the points $A, B, C, D$ respectively. The position vector of the centroid of the triangular face $BCD$ is $\frac{2}{3}(\vec{i}+\vec{j}+\vec{k})$. If $\alpha \vec{i}+\beta \vec{j}+\gamma \vec{k}$ is the position vector of the centroid of the tetrahedron $ABCD$, then find the value of $2 \alpha+\beta+\gamma$.

If the position vectors of $A$ and $B$ are $i + 3j - 7k$ and $5i - 2j + 4k$,then the direction cosine of $\overrightarrow{AB}$ along the $y$-axis is