So in my pre-calc class we have a pretty big test/exam on vectors today and I just want to make sure that I'm doing these word problems right.

1) I push a lawnmower with a force of 220 punds with the handle at an angle of 20degrees to the ground. How much force is actually being used in pushing the lawnmower forward?

The answer to this question would simply be 220(cos20deg) or ~206.732 pounds, correct?.

2) Two tugboats are pushing an ocean liner at angles of 17deg to the liner to the Northeast and Southeast. What is the resultant force on the ocean liner if both boats pull with a force of 500 tons?

I'm pretty sure I did this problem wrong, because I ended up with a force of 1000 tons. I wrote each vector as <500cos17 , 500sin17 > and <500cos-17 , 500sin-17 > , which gave me <956.3 , 292.4 > as the resultant vector.
The magnitude of the resultant vector = 1000 tons of force.

1) ok
2) it will be 500cos17 for both boats (aka just do that multiply by 2).
There will be no force in the other direction because they cancel out; North + South = 0.
500cos-17 doesn't exist either, they are both 500cos17 in the same direction of East.

It helps to draw pictures. But you got #1 right. Remember, you can always represent any vector as the sum of 2 vectors. So change the vector you are given into 2 vectors in the directions you are interested in. Here is a picture for #1. The green vector is the vector you were given, the 2 red vectors are a different representation of the green vector, and you can clearly see the answer:



#2
Same idea here. But clearly the 500 *sin(17) pointing South cancels the 500*sin(17) pointing North. Just leaving 2 vectors of 500*cos(17) pointing East. So adding them together gives 1000*cos(17) to the East.