Question

On a safari, a team of naturalists sets out toward a research station located 8.9 km away in a direction 42° north of east. After traveling in a straight line for 2.1 km, they stop and discover that they have been traveling 21° north of east, because their guide misread his compass. What is the direction (relative to due east) of the displacement vector now required to bring the team to the research station?

Answer 1

27.19°

Explanation:

According to the picture attached, we can find the distance between the two vectors using cosine law

`a^(2) =b^(2) +c^(2) -2ab*cosA\\a=\sqrt{b^(2) +c^(2) -2ab*cosA} \\\\a=\sqrt{2.1^(2) +8.9^(2) -2(2.1)(8.9)*cos21}\\a=6.98\\\\`

Then we can get C angle by applying one more time cosine law between a and b

`c^(2) =a^(2) +b^(2) -2ab*cosC\\\\c^(2) -a^(2) -b^(2)= -2ab*cosC\\\\(c^(2) -a^(2) -b^(2))/(-2ab)=cosC\\ \\CosC=(8.9^(2) -6.98^(2) -2.1^(2))/(-2*6.98*2.1)\\ \\CosC=-0.89\\\\ArcCos(-0.89)=C\\\\C=152.81`

We can see that the C angle is complement of the angle we are looking for, so we take away 180 degrees to get the answer

`180=C+?\\\\180-C=?\\\\180-152.81=C\\\\27.19=C`

27.19 degrees is our answer!