If $\vec{a}$ and $\vec{b}$ are two vectors such that $\vec{a}=2 \hat{i}+2 \hat{j}+p \hat{k}$, $|\vec{b}|=7$, $\vec{a} \cdot \vec{b}=4$ and $|\vec{a} \times \vec{b}|=5 \sqrt{17}$, then $p=$

If $|a| = 3, |b| = 1, |c| = 4$ and $a + b + c = 0,$ then $a \cdot b + b \cdot c + c \cdot a = $

If $\bar{a}, \bar{b}, \bar{c}$ are three unit vectors such that $|\bar{a}+\bar{b}+\bar{c}|=1$ and $\bar{b}$ is perpendicular to $\bar{c}$. If $\bar{a}$ makes angles $\alpha$ and $\beta$ with $\bar{b}$ and $\bar{c}$ respectively,then the value of $\cos \alpha+\cos \beta$ is:

If $a=\hat{i}+\hat{j}+t \hat{k}$ and $b=\hat{i}+2 \hat{j}+3 \hat{k}$, then the values of $t$ for which $(a+b)$ and $(a-b)$ are perpendicular are:

If the vectors $6i - 2j + 3k$,$2i + 3j - 6k$,and $3i + 6j - 2k$ are the position vectors of the vertices of a triangle,then the triangle is

If $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ are three vectors such that $\overrightarrow{a}=\overrightarrow{b}+\overrightarrow{c}$ and the angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is $\frac{\pi}{2}$, then: