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 sum of these vectors is

If $\vec{A}=\hat{i}+\hat{j}+3 \hat{k}, \vec{B}=-\hat{i}+\hat{j}+4 \hat{k}$ and $\vec{C}=2 \hat{i}-2 \hat{j}-8 \hat{k}$, then the angle between the vectors $\overrightarrow{\mathrm{P}}=\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}+\overrightarrow{\mathrm{C}}$ and $\overrightarrow{\mathrm{Q}}=(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}})$ is (in degree)

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 $\overrightarrow{\mathrm{C}}$ is

The angle subtended by the vector $A=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+12 \hat{\mathbf{k}}$ with the $X$-axis is