What is the projection vector of the vector $\vec{a} = (1, 1, 1)$ onto the vector $\vec{b} = (2, 2, 1)$?

Let $p=\hat{i}+2\hat{j}-\hat{k}$ and $q=2\hat{i}-\hat{j}+\hat{k}$. If $a$ and $b$ are two vectors such that $p=a-2b$ and $q=2a+b$, then the angle between $a$ and $b$ is

Assertion $(A)$: The scalar product of force $\vec{F}$ and displacement $\vec{r}$ is equal to the work done.
Reason $(R)$: The work done is not a scalar quantity.

If two vectors $\vec{a}$ and $\vec{b}$ are such that $|\vec{a}| = 2$, $|\vec{b}| = 3$ and $\vec{a} \cdot \vec{b} = 4$, then $|\vec{a} - \vec{b}| = . . . . . . $.

If $\bar{a}$ and $\bar{b}$ are two vectors such that $|\bar{a}|=5$, $|\bar{b}|=12$ and $|\bar{a}-\bar{b}|=13$, then $|2\bar{a}+\bar{b}|=$

The vector $\vec{a} + 3\vec{b}$ is perpendicular to $7\vec{a} - 5\vec{b}$ and the vector $\vec{a} - 4\vec{b}$ is perpendicular to $7\vec{a} - 2\vec{b}$. Then the angle between $\vec{a}$ and $\vec{b}$ is