Among three vectors,two vectors are equal in magnitude,and the magnitude of the third vector is $\sqrt{2}$ times the magnitude of the other two vectors. If $\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C} = 0$,find the angles between the vectors.

Vectors $\vec{A}$ and $\vec{B}$ make angles of $20^\circ$ and $110^\circ$ with the $x$-axis,respectively. The magnitudes of these vectors are $5 \ m$ and $12 \ m$,respectively. The angle that their resultant vector makes with the $x$-axis 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 $|\overrightarrow A| + |\overrightarrow B|$.

If $\overrightarrow A = 4\hat i - 3\hat j$ and $\overrightarrow B = 6\hat i + 8\hat j$,find the magnitude and direction of $\overrightarrow A + \overrightarrow B$.

$A$ vector of length $\ell$ is turned by an angle $\theta$. Find the change in the position vector of the tip.

Explain the triangle method (head-to-tail method) of vector addition.