If $P(3, 4, 5)$,$Q(4, 6, 3)$,$R(-1, 2, 4)$,and $S(1, 0, 5)$,then the projection of the vector $\vec{RS}$ on the vector $\vec{PQ}$ is:

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

Let $\bar{a}, \bar{b}, \bar{c}, \bar{d}$ be vectors such that $\bar{a} \times \bar{b} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\bar{c} \times \bar{d} = 3\hat{i} + 2\hat{j} + \lambda\hat{k}$. If $\begin{vmatrix} \bar{a} \cdot \bar{c} & \bar{b} \cdot \bar{c} \\ \bar{a} \cdot \bar{d} & \bar{b} \cdot \bar{d} \end{vmatrix} = 0$,then find the value of $\lambda$.

$a, b, c$ are three vectors such that $a + b + c = 0$,$|a| = 1, |b| = 2, |c| = 3$. Then $a \cdot b + b \cdot c + c \cdot a$ is equal to:

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

If $\overrightarrow{a}=-\hat{i}+\hat{j}+2 \hat{k}$, $\overrightarrow{b}=2 \hat{i}-\hat{j}-\hat{k}$ and $\overrightarrow{c}=-2 \hat{i}+\hat{j}+3 \hat{k}$, then the angle between $2 \overrightarrow{a}-\overrightarrow{c}$ and $\overrightarrow{a}+\overrightarrow{b}$ is