Farmer wants to Fence off his Four-Sided Plot with missing Side| Vector Addition and Subtraction| Analytical Method| College Physics| Problem 3.22

A farmer wants to fence off his four-sided plot of flat land. He measures the first three sides, shown as A, B, and C in the figure, and then correctly calculates the length and orientation of the fourth side D. What is his result?

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SOLUTION:

We know that the sum of the 4 vectors is zero. So, Vector D is calculated as

D=-A-B-C

We solve for the x-component first. The x-component of vector D is

D_x=-A_x-B_x-C_x

D_x=-\left(4.70\:km\right)cos\left(-7.50^{\circ} \right)-\left(2.48\:km\right)cos106^{\circ} -\left(3.02\:km\right)cos161^{\circ}

D_x=-1.12\:km

We solve for the y-component.

D_y=-A_y-B_y-C_y

D_y=-\left(4.70\:km\right)sin\left(-7.50^{\circ} \right)-\left(2.48\:km\right)sin106^{\circ} -\left(3.02\:km\right)sin161^{\circ}

D_y=-2.75\:km

Since we already know the x and y component of vector D, we can finally solve for the distance of vector D.

D=\sqrt{\left(D_x\right)^2+\left(D_y\right)^2}

D=\sqrt{\left(-1.12\:km\right)^2+\left(-2.75\:km\right)^2}

D=2.97\:km

The corresponding direction of vector D is

\theta =tan^{-1}\left|\frac{D_x}{D_y}\right|

\theta =tan^{-1}\left|\frac{1.12\:km}{2.75\:km}\right|

\theta =22.2^{\circ} \:W\:of\:S

 

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