Vector Displacement| Vector Addition and Subtraction: Graphical Method| Two-Dimensional Kinematics| College Physics Problem 3.6

Repeat the problem above, but reverse the order of the two legs of the walk; show that you get the same final result. That is, you first walk leg B , which is 20.0 m in a direction exactly 40º south of west, and then leg A , which is 12.0 m in a direction exactly 20º west of north. (This problem shows that A+B=B+A.)

SOLUTION:

First, we solve for the x and y components of vectors A and B.

For vector A, the components are:

A_x=-\left(12.0\:m\right)sin\:\left(20^{\circ} \right)=-4.1042\:m

A_y=\left(12.0\:m\right)cos\left(20^{\circ} \right)=11.2763\:m

For vector B, the components are:

B_x=-\left(20\:m\right)cos\left(40^{\circ} \right)=-15.3209\:m

B_y=-\left(20\:m\right)sin\:\left(40^{\circ} \right)=-12.8557\:m

Since we know the components, we can now solve for the resultant vector R by adding the components together. 

The resultant of the x-components is:

R_x=A_x+B_x=-4.1042+\left(-15.3209\right)=-19.4251\:m

The resultant of the y-components is:

R_y=A_y+B_y=11.2763+\left(-12.8557\right)=-1.5794\:m

The magnitude of the resultant can be solved using the Pythagorean Theorem. That is:

R=\sqrt{\left(R_x\right)^2+\left(R_y\right)^2}=\sqrt{\left(-19.4251\right)^2+\left(-1.5794\right)^2}=19.4892\:m

The compass direction is

\theta =tan^{-1}\left|\frac{R_y}{R_x}\right|=tan^{-1}\left(\frac{1.5794}{19.4251}\right)=4.6483^{\circ}

The compass direction is 4.6483^{\circ} \:South\:of\:West.

No matter what is the order of the vectors we are adding, the result is still the same. This proves that \vec{A}+\vec{B}=\vec{B}+\vec{A}.

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