Vector addition using components

[chocolaterie]

Homework Statement

You will be hiking to a lake with some of your friends. the map says you will travel 1.6 mi north then 2.2 mi in a direction 35 degrees east of north, then finally 1.1 mi in a direction 15 degrees north of east. how far will you be from where you started and what direction will you be from your starting point?

The Attempt at a Solution

I have no idea where to start. I drew a graph with 1.6 mi on the y-axis going north.

 

[mikelepore]

When you have a right triangle, remember the definition of the sine and cosine of an angle. Try to use that to write down the x component and y component of each part of the motion. Add up all the x components, and in a separate step add up all the y components.

 

[tomcue]

Then, I added 2.2 mi in a direction 35 degrees east, but I'm not sure how to find the final distance and direction.I would suggest using vector addition using components to solve this problem. This method involves breaking down each vector into its horizontal and vertical components and then adding them together to find the resultant vector. In this case, the first vector of 1.6 mi north would have a vertical component of 1.6 mi and a horizontal component of 0 mi. The second vector of 2.2 mi in a direction 35 degrees east of north would have a vertical component of 2.2*cos(35) mi and a horizontal component of 2.2*sin(35) mi. Similarly, the third vector of 1.1 mi in a direction 15 degrees north of east would have a vertical component of 1.1*cos(15) mi and a horizontal component of 1.1*sin(15) mi.

Now, we can add all the vertical components and all the horizontal components separately to get the final resultant vector. The magnitude of the resultant vector can be found using the Pythagorean theorem (a^2 + b^2 = c^2) and the direction can be found using trigonometric functions (tan^-1(b/a)). The final distance from the starting point would be the magnitude of the resultant vector and the direction would be the angle formed by the resultant vector with the x-axis.

Using this method, the final distance from the starting point would be approximately 2.87 mi and the direction would be approximately 30.7 degrees east of north. I hope this helps in solving the problem.