If $|\vec{a}| = \sqrt{26}$,$|\vec{b}| = 7$,and $|\vec{a} \times \vec{b}| = 35$,find $\vec{a} \cdot \vec{b}$.

$\vec{A}$ is a vector quantity such that $|\vec{A}| =$ nonzero constant. Which of the following expressions is true for $\vec{A}$?

The dot product of unit vectors $\hat{n}_1$ and $\hat{n}_2$ that are parallel to $5 \hat{i}+12 \hat{j}$ and $3 \hat{i}+4 \hat{j}$ respectively is

Find the unit vector perpendicular to $\vec{A}$ and $\vec{B}$ where $\vec{A} = \hat{i} - 2\hat{j} + \hat{k}$ and $\vec{B} = \hat{i} + 2\hat{j}$.

$A$ force $F$ applied on a body is written as $F = (\hat{n} \cdot F) \hat{n} + G$,where $\hat{n}$ is a unit vector. The vector $G$ is equal to

If $F_1$ and $F_2$ are two vectors of equal magnitudes $F$ such that $|F_1 \cdot F_2| = |F_1 \times F_2|$,then $|F_1 + F_2|$ is equal to