The angle made by the resultant vector of two vectors $2 \hat{i}+3 \hat{j}+4 \hat{k}$ and $2 \hat{i}-7 \hat{j}-4 \hat{k}$ with the $x$-axis is (in $^{\circ}$)

The magnitude of the $X$ and $Y$ components of $\vec{A}$ are $7$ and $6$ respectively. Also,the magnitude of the $X$ and $Y$ components of $\vec{A} + \vec{B}$ are $11$ and $9$ respectively. What is the magnitude of $\vec{B}$?

If three vectors have equal magnitude,i.e.,$A = B = C$,then the angle between $\vec{A}$ and $\vec{C}$ is $\alpha$. If $\vec{A} + \vec{B} + \vec{C} = 0$,then the angle between $\vec{A}$ and $\vec{C}$ is $\beta$. Find the ratio $\frac{\alpha}{\beta}$.

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

If the sum of two unit vectors is a unit vector,find the magnitude of their difference.

$A$ football player is moving southward and suddenly turns eastward with the same speed to avoid an opponent. The force that acts on the player while turning is: