If $(\vec{a}+3 \vec{b})$ is perpendicular to $(7 \vec{a}-5 \vec{b})$ and $(\vec{a}-4 \vec{b})$ is perpendicular to $(7 \vec{a}-2 \vec{b})$,then the angle between $\vec{a}$ and $\vec{b}$ (in degrees) is $......$

Magnitudes of vectors $\vec{a}, \vec{b}, \vec{c}$ are $3, 4, 5$ respectively. If $\vec{a}$ and $\vec{b} + \vec{c}$,$\vec{b}$ and $\vec{c} + \vec{a}$,and $\vec{c}$ and $\vec{a} + \vec{b}$ are mutually perpendicular,then the magnitude of $\vec{a} + \vec{b} + \vec{c}$ is:

For any two vectors $\vec{a}$ and $\vec{b}$,$|\vec{a}| |\vec{b}|$ . . . . . . $|\vec{a} \cdot \vec{b}|$.

In $\triangle OAB$, $O(0, 0, 0)$, $A(6, 2, -3)$ and $B(4, 0, 3)$ are the vertices. Let $\vec{a}$ and $\vec{b}$ be position vectors of points $A$ and $B$ respectively. If $OM$ is the projection of $\vec{a}$ on $\vec{b}$, then the length $l(AM)$ is equal to...

If $a, b, c$ are three mutually perpendicular vectors such that the magnitudes of $b$ and $c$ are $1/2$ times and $\sqrt{3}/2$ times that of $a$, respectively, then the angle between the vectors $a+b+c$ and $b$ is

The scalar product of the vector $\hat{i}+\hat{j}+\hat{k}$ with a unit vector along the sum of vectors $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\lambda \hat{i}+2 \hat{j}+3 \hat{k}$ is equal to $1$. Find the value of $\lambda$.