You leave your friend behind on the shore and you travel 3 m

You leave your friend behind on the shore and you travel 3 miles due east in your boat. Then you travel 2 miles northeast. Then you travel 1 mile due north.

Your friend can see you at a distance of *blank* miles and at a bearing of *blank* degrees.
Enter your answers with at least 3 digits beyond the decimal point.

Hint: Carefully draw a picture. Use the Laws of Sines and Cosines.

Solution

This is vector addition.

You start at the origin (0,0)
3 mi due east --> (3,0)

2 mi northeast (45 degrees).
Decompose this into x and y
x is 2sqrt(2)/2 = 1.414
y is 2sqrt(2)/2 = 1.414

1 mi due north (in the y direction)

Now add the components

sum of x\'s = 3+1.414+0 = 4.414
sum of y\'s = 0+1.414+1 = 2.414

The resultant vector starts at (0,0) and ends at (4.414,2.414)
distance = sqrt[(4.414)^2 + (2.414)^2] = 5.031 mi

tan(theta) = 2.414/4.414 (opposite over adjacent)

theta = 28.7 degrees north of east