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 $^{\circ}$)

Three vectors are expressed as $\vec{a} = 4\hat{i} - \hat{j}$,$\vec{b} = -3\hat{i} + 2\hat{j}$,and $\vec{c} = -\hat{k}$. The unit vector along the direction of the sum of these vectors is:

Given that $A = B = C$. If $\vec A + \vec B = \vec C$,then the angle between $\vec A$ and $\vec C$ is $\theta_1$. If $\vec A + \vec B + \vec C = 0$,then the angle between $\vec A$ and $\vec C$ is $\theta_2$. What is the relation between $\theta_1$ and $\theta_2$?

The resultant of $\vec{A} + \vec{B}$ is $\vec{R}_1$. On reversing the vector $\vec{B}$,the resultant becomes $\vec{R}_2$. What is the value of $R_1^2 + R_2^2$?

Two forces,each numerically equal to $10$ $dyne$,are acting as shown in the adjoining figure. The magnitude of the resultant force is ......... $dyne$.

Given $\vec{A}=(2 \hat{i}-3 \hat{j}+\hat{k})$,$\vec{B}=(3 \hat{i}+\hat{j}-2 \hat{k})$ and $\vec{C}=(3 \hat{i}+2 \hat{j}+\hat{k})$. The value of $(\vec{A}+\vec{B}) \cdot \vec{C}$ will be