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Hilary and Dale leave camp and go for a long hike. After going 7 km due east, they turn and go another 8 km in the direction 60 degrees north of east. They plan to return along a straight path. How far from camp are they at this point? Use an angle to describe the direction that Hilary and Dale should follow to reach their camp.

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Answer:

Step-by-step explanation:

Given

First they travel 7 km east , then they go 8 km in direction 60^{\circ} north of east

Position vector of first point

[tex]r_1=7\hat{i}[/tex]

[tex]r_{21}=8cos60\hat{i}+8sin60\hat{j}[/tex]

[tex]r_{21}=4\hat{i}+6.92\hat{j}[/tex]

Position vector of [tex]r_2[/tex] w.r.t to origin

[tex]r_2=r_{21}+r_1[/tex]

[tex]r_2=4\hat{i}+6.92\hat{j}+7\hat{i}[/tex]

[tex]r_2=11\hat{i}+6.92\hat{j}[/tex]

magnitude of [tex]r_2[/tex] will give the distance they are from original point

[tex]|r_2|=\sqrt{6.92^2+11^2}=13[/tex]

direction

[tex]tan\theta =\frac{y}{x}=\frac{6.92}{11}[/tex]

[tex]\theta =32.17^{\circ}[/tex] North of east

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