The resultant of two vectors $\vec{A}$ and $\vec{B}$ is $\vec{C}$. If the magnitude of $\vec{B}$ is doubled,the new resultant vector becomes perpendicular to $\vec{A}$. Then,the magnitude of $\vec{C}$ is:

Two forces of magnitude $3\;N$ and $4\;N$ respectively are acting on a body. Calculate the resultant force if the angle between them is $0^{\circ}$. (in $;N$)

Two vectors $\vec A$ and $\vec B$ have equal magnitudes. The magnitude of $(\vec A + \vec B)$ is $n$ times the magnitude of $(\vec A - \vec B)$. The angle between $\vec A$ and $\vec B$ is

If vectors $\overrightarrow P, \overrightarrow Q$ and $\overrightarrow R$ have magnitudes $5, 12$ and $13$ units respectively and $\overrightarrow P + \overrightarrow Q = \overrightarrow R$,then the angle between $\overrightarrow Q$ and $\overrightarrow R$ is

Let $\overrightarrow C = \overrightarrow A + \overrightarrow B$. Which of the following statements are correct?
$(A)$ It is possible to have $|\overrightarrow C| < |\overrightarrow A|$ and $|\overrightarrow C| < |\overrightarrow B|$.
$(B)$ $|\overrightarrow C|$ is always greater than $|\overrightarrow A|$.
$(C)$ $|\overrightarrow C|$ may be equal to $|\overrightarrow A| + |\overrightarrow B|$.
$(D)$ $|\overrightarrow C|$ is never equal to $|\overrightarrow A| + |\overrightarrow B|$.

Given $A = 2 \hat{i} + \hat{j}$,$B = 3 \hat{j} - \hat{k}$,and $C = 6 \hat{i} - 2 \hat{k}$. The value of $A - 2B + 3C$ is: