Determine the sum of vectors A , B , and C , each possessing a...

The vectors can be expressed in Cartesian coordinates as follows:

\vec{A} = 100(\cos 45^{}^{{}^{\circ}} \hat{i} + \sin 45^{}^{{}^{\circ}} \hat{j})

\vec{B} = 100(\cos 135^{}^{{}^{\circ}} \hat{i} + \sin 135^{}^{{}^{\circ}} \hat{j}) = 100(-\cos 45^{}^{{}^{\circ}} \hat{i} + \sin 45^{}^{{}^{\circ}} \hat{j})

\vec{C} = 100(\cos 315^{}^{{}^{\circ}} \hat{i} + \sin 315^{}^{{}^{\circ}} \hat{j}) = 100(\cos 45^{}^{{}^{\circ}} \hat{i} - \sin 45^{}^{{}^{\circ}} \hat{j})

Observing vectors $\vec{B}$ and $\vec{C}$, their angles differ by 315^{}^{{}^{\circ}} - 135^{}^{{}^{\circ}} = 180^{}^{{}^{\circ}}. Since they have equal magnitudes and are in opposite directions, their sum is zero:

$\vec{B} + \vec{C} = \vec{0}$

The resultant of the three vectors is:

$\vec{R} = \vec{A} + \vec{B} + \vec{C} = \vec{A} + \vec{0} = \vec{A}$

Therefore, the resultant vector is equal to $\vec{A}$, which has a magnitude of $100$ units and is inclined at 45^{}^{{}^{\circ}} with the X-axis.