Let $\vec{a}, \vec{b}, \vec{c}$ be vectors of lengths $3, 4, 5$ respectively. Let $\vec{a}$ be perpendicular to $\vec{b}+\vec{c}$,$\vec{b}$ be perpendicular to $\vec{c}+\vec{a}$,and $\vec{c}$ be perpendicular to $\vec{a}+\vec{b}$. Then the length of vector $\vec{a}+\vec{b}+\vec{c}$ is:

Three vectors $\vec a, \vec b, \vec c$ are inclined at an acute angle with each other such that $|\vec a| = 2, |\vec b| = 3, |\vec c| = 9$ and the lengths of the projections of $\vec a$ on $\vec b$,$\vec b$ on $\vec c$,and $\vec c$ on $\vec a$ respectively are in geometric progression. If the angle between $\vec a$ and $\vec b$ is $\frac{5\pi}{12}$ and the angle between $\vec c$ and $\vec a$ is $\frac{\pi}{12}$,then the angle between $\vec b$ and $\vec c$ is:

Suppose $\overrightarrow{a}=\lambda \hat{i}-7 \hat{j}+3 \hat{k}$ and $\overrightarrow{b}=\lambda \hat{i}+\hat{j}+2 \lambda \hat{k}$. If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is greater than $90^{\circ}$, then $\lambda$ satisfies the inequality:

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be vectors of magnitude $2, 3$ and $4$ 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 magnitude of $\overline{a}+\overline{b}+\overline{c}$ is equal to

If $a = i + j$ and $b = 2i - k$ are two vectors, then the point of intersection of two lines $r \times a = b \times a$ and $r \times b = a \times b$ 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 . . . . . . .