If $\vec{a}$ and $\vec{b}$ are two vectors such that $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} < 0$ and $|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|$,then the angle between the vectors $\vec{a}$ and $\vec{b}$ is

Let $a = 2i + j + k$ and $b = i + 2j - k$. If a unit vector $c$ is coplanar with $a$ and $b$,and $c$ is perpendicular to $a$,then $c$ is:

If $\overrightarrow{a} = \hat{i} + 2\hat{k}$,$\overrightarrow{b} = \hat{i} + \hat{j} + \hat{k}$,and $\overrightarrow{c} = 7\hat{i} - 3\hat{j} + 4\hat{k}$,such that $\overrightarrow{r} \times \overrightarrow{b} + \overrightarrow{b} \times \overrightarrow{c} = \overrightarrow{0}$ and $\overrightarrow{r} \cdot \overrightarrow{a} = 0$,then $\overrightarrow{r} \cdot \overrightarrow{c}$ is equal to:

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|2 \vec{a}+3 \vec{b}|=|3 \vec{a}+\vec{b}|$ and the angle between $\vec{a}$ and $\vec{b}$ is $60^{\circ}$. If $\frac{1}{8} \vec{a}$ is a unit vector,then $|\vec{b}|$ is equal to :

Let the position vectors of the points $P, Q, R$ and $S$ be $\vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}$,$\vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}$,$\vec{c}=\frac{17}{5} \hat{i}+\frac{16}{5} \hat{j}+7 \hat{k}$ and $\vec{d}=2 \hat{i}+\hat{j}+\hat{k}$,respectively. Then which of the following statements is true?

The vector $2i + j - k$ is perpendicular to $i - 4j + \lambda k,$ if $\lambda = $