The projection of $\bar{a} = \hat{i} - 2\hat{j} + \hat{k}$ on $\bar{b} = 2\hat{i} - \hat{j} + \hat{k}$ is

Let $\vec{a}=\hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}$ be a vector such that $\vec{a} \times \vec{b}=2 \hat{i}-\hat{k}$ and $\vec{a} \cdot \vec{b}=3$. Then the projection of $\vec{b}$ on the vector $\vec{a}-\vec{b}$ is :-

Let $\vec{a}=\hat{i}$ and $\vec{b}=\hat{j}$. The point of intersection of the lines $\vec{r} \times \vec{a}=\vec{b} \times \vec{a}$ and $\vec{r} \times \vec{b}=\vec{a} \times \vec{b}$ is:

The vectors $(a \cdot b) c$ and $(a \cdot c) b$ are:

The projection of the line segment joining the points $P(1, -1, 3)$ and $Q(2, -4, 11)$ on the line joining the points $A(-1, 2, 3)$ and $B(3, -2, 10)$ is

If the vectors $a\,i - 2j + 3k$ and $3i + 6j - 5k$ are perpendicular to each other,then $a$ is given by