If $\vec{A} = 2\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{B} = \hat{i} + 4\hat{j} - 2\hat{k}$, and $\vec{C}$ is a vector perpendicular to both $\vec{A}$ and $\vec{B}$ such that $\vec{C} \cdot (\hat{i} + \hat{j} - \hat{k}) = 3$, find the magnitude of $\vec{C}$.

Two vectors $\vec{A}$ and $\vec{B}$ satisfy the condition $|\vec{A} + \vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2$. What is the angle between $\vec{A}$ and $\vec{B}$?

If the dot product of two non-zero vectors is zero, what can be concluded about the angle between them?

The X and Y components of vector ${\rm{\vec A}}$ have numerical values 6 and 6 respectively and that of $\left( {{\rm{\vec A}} + {\rm{\vec B}}} \right)$ have numerical values 10 and 9. What is the numerical value of ${{\rm{\vec B}}}$ ?

If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors, the angle between these vectors is:

A force is inclined at ${60^0}$ to the horizontal. If its rectangular component in the horizontal direction is 50 N, then magnitude of the force in the vertical direction is

What is the angle between ${\rm{\vec P}}\,and\,{\rm{\vec Q}}$. The resultant of $\left( {{\rm{\vec P}} + {\rm{\vec Q}}} \right)\,and\,\left( {{\rm{\vec P}} - {\rm{\vec Q}}} \right)?$

If the resultant of the vectors $\left( {\hat i + 2\hat j - \hat k} \right),\left( {\hat i - \hat j + 2\hat k} \right)$ and ${\rm{\vec C}}$ is a unit vector along the y-direction, then ${\rm{\vec C}}$ is

If the magnitude of the sum of two vectors is equal to the magnitude of their difference, what is the angle between the two vectors?

For two vectors $\vec{A}$ and $\vec{B}$, if $|\vec{A} + \vec{B}| = |\vec{A} - \vec{B}|$, what is the angle between $\vec{A}$ and $\vec{B}$?

If three unit vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ satisfy $\vec{a} + \vec{b} + \sqrt{3}\vec{c} = \vec{0}$, then the angle between $\vec{a}$ and $\vec{b}$ is: