The values of $x$ for which the angle between the vectors $x^2 \hat{i} + 2 x \hat{j} + \hat{k}$ and $\hat{i} - 2 \hat{j} + x \hat{k}$ is obtuse,lie in the interval

If $a$ and $b$ are mutually perpendicular vectors,then $(a + b)^2 = $

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}|=3, |\vec{b}|=4, |\vec{c}|=5$ and each one of them is perpendicular to the sum of the other two. Find $|\vec{a}+\vec{b}+\vec{c}|$.

If $\vec{a}, \vec{b}$ and $\vec{c}$ are vectors with magnitudes $2, 3$ and $4$ respectively,then the best upper bound of $|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2$ among the given values is

$a, b$ and $c$ are three vectors with magnitudes $|a| = 4, |b| = 4, |c| = 2$ such that $a$ is perpendicular to $(b + c)$,$b$ is perpendicular to $(c + a)$,and $c$ is perpendicular to $(a + b)$. It follows that $|a + b + c|$ is equal to

If $\overline{p}=2 \hat{i}+\hat{k}$,$\overline{q}=\hat{i}+\hat{j}+\hat{k}$,$\overline{r}=4 \hat{i}-3 \hat{j}+7 \hat{k}$ and a vector $\overline{m}$ is such that $\overline{m} \times \overline{q}=\overline{r} \times \overline{q}$ and $\overline{m} \cdot \overline{p}=0$,then $\overline{m} = \dots$