The Resultant Vector

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All vectors repositioned tail at origin, and angle from 0°.

Then you can simply take the angle from the origin with cosine and sine and sum them.

$$ \vec{R}=\left\langle \sum x,\sum y\right\rangle $$

$$ |\vec{R}|=\sqrt{\left[\sum x\right]^2+\left[\sum y\right]^2} $$

$$ \theta=\arctan\left[\frac{\sum y}{\sum x}\right] $$

NOTE* Depending on which quadrant you are in, you may have to add 90 or 180 degrees.

Then, the Resultant vector will be,

$$ \vec{A} = \langle |\vec{A}|\cos\alpha,|\vec{A}|\sin\alpha\rangle, \qquad \vec{B} = \langle |\vec{B}|\cos\beta,|\vec{B}|\sin\beta\rangle, \qquad \vec{C} = \langle |\vec{C}|\cos\gamma,|\vec{C}|\sin\gamma\rangle. $$

$$ R_x = |\vec{A}|\cos\alpha + |\vec{B}|\cos\beta + |\vec{C}|\cos\gamma = A_x+B_x+C_x, $$

$$ R_y = |\vec{A}|\sin\alpha + |\vec{B}|\sin\beta + |\vec{C}|\sin\gamma = A_y+B_y+C_y, $$

$$ \vec{R} = \langle R_x,R_y\rangle = \langle A_x+B_x+C_x,A_y+B_y+C_y\rangle, $$

$$ |\vec{R}| = \sqrt{ [A_x+B_x+C_x]^2 + [A_y+B_y+C_y]^2 }, $$

$$ \tau = \arctan \left[ \frac{A_y+B_y+C_y} {A_x+B_x+C_x} \right]. $$

The resultant vector that we found is the same as finding the net force or work … as you progress deeper into the course.

• The first 4 or 5 chapters of the book lead up to learning how to work with all the vectors. Then, it basically repeats itself just calling the resultant something like net force or total work, …

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