How to Add Vectors on a Grid
You can use the components of vectors to add vectors together using a grid. Doing so reduces the problem of adding vectors to a simple combination of adding numbers together, which is very useful when you solve physics problems.
Take a look at the vector addition problem A + B in the above figure. Now that you have the vectors plotted on a graph, you can see how easy vector addition really is. If the measurements in the figure are in meters, that means vector A is 5 meters to the right and 1 meter up, and vector B is 1 meter to the right and 4 meters up. To add them for the result, vector C, you add the horizontal parts together and the vertical parts together.
The resultant vector, C, ends up being 6 meters to the right and 5 meters up. You can see what that looks like in the figure: To get the horizontal part of the sum, you add the horizontal part of A (5 meters) to the horizontal part of B (1 meter). To get the vertical part of the sum, C, you just add the vertical part of A (1 meter) to the vertical part of B (4 meters).
If vector addition still seems cloudy, you can use a notation that was invented for vectors to help physicists and For Dummies readers keep it straight. Because A is 5 meters to the right (the positive x-axis direction) and 1 up (the positive y-axis direction), you can express it with (x, y) coordinates like this:
A = (5, 1)
And because B is 1 meter to the right and 4 up, you can express it with (x, y) coordinates like this:
B = (1, 4)
Having a notation is great, because it makes vector addition totally simple. To add two vectors together, you just add their x and y parts, respectively, to get the x and y parts of the result:
A (5, 1) + B (1, 4) = C (6, 5)
The whole secret of vector addition is breaking each vector up into its x and y parts and then adding those separately to get the resultant vector’s x and y parts. Nothing to it. Now you can get as numerical as you like, because you’re just adding or subtracting numbers. Getting those x and y parts can take a little work, but it’s a necessary step. And when you have those parts, you’re home free.
Here’s a real-world example: Assume you’re looking for a hotel that’s 20 miles due north and then 20 miles due east. What’s the vector that points at the hotel from your starting location? Taking your coordinate info into account, this is an easy problem. Say that the east direction is along the positive x-axis and that north is along the positive y-axis. Step 1 of your travel directions is 20 miles due north, and Step 2 is 20 miles due east. You can write the problem in vector notation like this (east [positive x], north [positive y]):
Step 1: (0, 20)
Step 2: (20, 0)
To add these two vectors together, add the coordinates:
(0, 20) + (20, 0) = (20, 20)
The resultant vector is (20, 20). It points from your starting point directly to the hotel.