Vectors
In physics we often need to describe direction as well as size.For example, two forces F_{1} and F_{2} may both be 100 N and acting on a crocodile, but the crocodile's experience will be very different depending on whether the forces are both pointing north or one north and one south (Figure 22).
In the former case he gets stretched, and in the latter case he goes flying. To describe forces we need to specify size and direction. That is, we need to use vectors. Force is a vector. We denote vectors in diagrams by arrows, the length of the arrow showing the size of the vector and the direction of the arrow showing its direction.
We can add vectors by the tiptotail method. We leave the first vector fixed, and move the second vector so its tail is at the first vector's tip. If there are other vectors, then each vector gets added to the previous tip. The sum is the vector pointing from the first tail to the last tip.
Figure 22
Example 1:
For the crocodiles mentioned before, if the vectors (both 100 N) both point north, then the sum is a force of 200 N pointing north (Figure 23, where the sum is shown dashed).
If one vector points north and the other south, then the first tail coincides with the last tip and the sum is zero (Figure 23).
Figure 23
A crocodile has three forces acting on him: a 100N force north, a 100N force east, and a 100N force southwest. What is the direction of the net force (that is, total force)?
We DRAW A DIAGRAM (Figure 24). The sum is a vector pointing northeast, about 40 N.
Figure 24
Note that, when you add vectors, the magnitude of the sum is equal to, at most, the sum of the individual magnitudes (and that only if they are pointing in the same direction). For instance, if three vectors of 100 N are acting on a crocodile, the sum can be anything from 0 N to 300 N, but no greater.
For MCAT problems, vector addition need not get more sophisticated than this. It is useful to keep in mind the Pythagorean theorem and elementary trigonometry.
A force of 4 N is acting to the north on a rock and a force of 3 N is acting to the east.

What is the magnitude of the total force?

What is cosÏ†, if Ï† is the deviation from north of the direction of the total force?
We DRAW A DIAGRAM (Figure 25). There is a right triangle, so we can write
Figure 25
F_{sum} = 5 N
Also we write
cosÏ† = 4N/5N = 0.8
If your trigonometry is rusty, now is a good time to relearn the definitions of sine, cosine, and tangent.
A vector is denoted by a halfarrow on top of a letter, , for example.