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? 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 $\theta =tan^{-1}\left|\frac{D_x}{D_y}\right|$ $\theta =tan^{-1}\left|\frac{1.12\:km}{2.75\:km}\right|$ $\theta =22.2^{\circ} \:W\:of\:S$