Now that we have some comfort with vectors, in being able to identify them, understanding some of their properties, and breaking them into components, there is one last large piece of the puzzle to having a solid foundation of vectors and vector operations. Because vectors are not simply one number, many of our mathematical tools do not apply. How do you add a coordinate point? How do you subtract it? Even worse, how do you multiply it? These answers will be found in this chapter, as well as some very useful vector properties, and their associated proofs. So let’s start with the usual first: addition. There are two methods of adding vectors, one mathematical, the other graphical. First the math: consider you have two vectors, [vx,vy,vz] and [wx,wy,wz]. The way to add these vectors to produce a vector is highly intuitive: add each component to create the component of your final vector, i.e. [vx+wx,vy+wy,vz+wz]. Finished. Nothing more to it. This actually gives us a very valuable result: vector addition is commutative, i.e. it does not matter which order you add the vectors. Likewise, subtraction gives a very similar result:
This, as you might expect, means that vector subtraction is not commutative, i.e. the order of vectors matters.
Consider a more visual representation of vector addition: Imagine that you are standing at the origin of a coordinate axis, like graph paper, and decide to go for a little walk. You first decide to tightrope your way down the positive x-axis and you move 4 meters. You then turn 90 degrees to your left and walk 3 meters up, parallel with the y-axis, but 5 meters displaced. What is your overall motion, or displacement? We’re considering two different displacement vectors in this scenario. The first is the tightrope down the x-axis, which moved us from position (0,0) to (4,0). This is equivalent to a displacement vector [4,0]. Next, the walk parallel with the y-axis. This is a displacement of [0,3]. The vector [0,3] is akin to simply having walked up the y-axis first. This is one of the wonderful properties of vectors. When you think of them as arrows, you can slide them around wherever you want, and they retain the same properties. That’s what we’re going to do here. So, our overall displacement will be the sum of each increment that we’ve walked. This means
In this example, it’s very easy to see how the addition works in each direction, as the motions are both conveniently along axes, giving us the easy task of adding an integer and zero.
Q: How come to calculate the displacement I don’t implement the Pythagorean Theorem?
A: If you wanted to calculate the magnitude of displacement, you would do exactly that, however the angle and magnitude of displacement are retained in the value [4,3]. You could also say that the overall displacement was 5m, at an angle of ~37 degrees w/r/t the x-axis. Mathematically, it is usually simpler to keep the entire displacement in terms of the overall x, y and sometimes z components. The takeaway from the example: in order to add vectors graphically,
1) “Lay” your first vector.
2) Place the “tail” of the second vector at the “tip” of the first.
3) Repeat for any following vectors that need to be added.
4) Connect the tail of the first vector to the tip of the last vector. This is your resultant vector.
