The values of $x$ for which the angle between the vectors $\bar{a} = 2x^2 \hat{i} + 4x \hat{j} + \hat{k}$ and $\bar{b} = 7 \hat{i} - 2 \hat{j} + x \hat{k}$ is obtuse,are

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 $p=$

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that $|(\hat{a}+\hat{b})+2(\hat{a} \times \hat{b})|=2$. If $\theta \in(0, \pi)$ is the angle between $\hat{a}$ and $\hat{b}$,then among the statements:
$(S_{1})$: $2|\hat{a} \times \hat{b}|=|\hat{a}-\hat{b}|$
$(S_{2})$: The projection of $\hat{a}$ on $(\hat{a}+\hat{b})$ is $\frac{1}{2}$

If $\theta$ is the angle between any two vectors $\vec{a}$ and $\vec{b},$ then $|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$ when $\theta$ is equal to

Let $\bar{a} = 4\bar{i} + 3\bar{j}$ and $\bar{b}$ be two perpendicular vectors in the $XOY$-plane. $A$ vector $\bar{c}$ in the same plane and having projections $1$ and $2$ respectively on $\bar{a}$ and $\bar{b}$ is

If $|\bar{a}|=\sqrt{26}$,$|\bar{b}|=7$,and $|\bar{a} \times \bar{b}|=35$,then $\bar{a} \cdot \bar{b}=$