If $\vec{\lambda}$ is a unit vector perpendicular to the plane of vectors $\vec{a}$ and $\vec{b}$,and the angle between them is $\theta$,then $\vec{a} \cdot \vec{b}$ will be:

If $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are unit vectors such that $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$,then the value of $3 \overrightarrow{a} \cdot \overrightarrow{b}+2 \overrightarrow{b} \cdot \overrightarrow{c}+\overrightarrow{c} \cdot \overrightarrow{a}$ is

Let $\vec{a}$ and $\vec{b}$ be two vectors. Let $|\vec{a}|=1, |\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$,then the value of $\vec{b} \cdot \vec{c}$ is

For vectors $\vec{a}, \vec{b}$ and $\vec{c}$,if $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ and $|\vec{a}|=2, |\vec{b}|=3, |\vec{c}|=5$,then the value of $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ is . . . . . . .

Let $\overline{A}, \overline{B}, \overline{C}$ be vectors of lengths $3$ units,$4$ units,and $5$ units respectively. If $\overline{A}$ is perpendicular to $\overline{B}+\overline{C}$,$\overline{B}$ is perpendicular to $\overline{C}+\overline{A}$,and $\overline{C}$ is perpendicular to $\overline{A}+\overline{B}$,then the length of vector $\overline{A}+\overline{B}+\overline{C}$ is

If $\vec{a}$ and $\vec{b}$ are unit vectors,then the angle between $\vec{a}$ and $\vec{b}$ for $\sqrt{3}\vec{a}-\vec{b}$ to be a unit vector is (in $^{\circ}$)