Video Transcript
Which of the vectors 𝐏, 𝐐, 𝐑, 𝐒, and 𝐓 shown in the diagram is equal to 𝐀 plus 𝐁 plus 𝐂?
The diagram referred to is this set of Cartesian axes with several vectors all represented as arrows. We are particularly interested in the vectors 𝐀, 𝐁, and 𝐂. And what we are tasked with doing is identifying which of the other vectors in the diagram is equal to the sum of these three vectors. When we represent a vector with an arrow, the pointy end is called the head and the other end is called the tail. To add a second vector, we simply draw that vector with its tail at the head of this first one. The sum of these two vectors is then the vector whose tail is at the tail of the first vector and whose head is at the head of the second vector.
However, for our question, we’re looking for a sum of three vectors, not just two. Luckily, vector addition is associative. This means that we can group the various parts of the calculation however we want without changing the final answer. So say we are adding three vectors 𝐔, 𝐕, and 𝐖. To add all three of these vectors together, we can first add the two vectors 𝐔 and 𝐕 and then add 𝐖.
As we can see from our diagram though, 𝐔 plus 𝐕 is just another single vector, so 𝐔 plus 𝐕 in parentheses plus 𝐖 is just a sum of this new vector 𝐔 plus 𝐕 with the third vector 𝐖. And we know how to do that. We just draw 𝐖 with its tail at the head of 𝐔 plus 𝐕. Now we draw the vector whose tail is at the tail of 𝐔 plus 𝐕 and whose head is at the head of 𝐖. And this gives us the sum of the three vectors 𝐔 plus 𝐕 plus 𝐖. Note we could actually erase the intermediate step of calculating 𝐔 plus 𝐕 and still get the same answer.
So we can calculate the sum 𝐔 plus 𝐕 plus 𝐖 by drawing each arrow with its tail at the head of the previous arrow and then connecting the remaining tail and the remaining head to form the final vector. In fact, this works for any number of vectors, and we will always get the same answer regardless of the order that we use when drawing the vectors. And this works because vector addition is both commutative and associative.
So to find the sum we are looking for, we just need to redraw 𝐀, 𝐁, and 𝐂 with their heads and tails touching. Let’s start by redrawing 𝐁 with its tail at the head of 𝐀. The arrow for 𝐁 extends two units to the right and three units upward. Drawing an arrow with its tail at the head of 𝐀 and extending two units to the right and three units upward brings us to this point here. Now we need to add the arrow for 𝐂. This arrow extends four units to the left and one unit upward. Connecting this arrow to the two that we’ve already drawn brings us from the previous head to this point here, which is the head of the vector labeled 𝐐.
Now, to find the sum that we’re looking for, we need to find the vector that has its tail at the tail of 𝐀 and its head at the head of 𝐂. And we see that 𝐐 indeed has its head at the head of 𝐂. And since both 𝐀 and 𝐐 have their tails at the origin, 𝐐 also has its tail at the tail of 𝐀. So because when we connect 𝐀, 𝐁, and 𝐂 from head to tail, 𝐐 connects the remaining tail to the remaining head, we conclude that 𝐐 is equal to 𝐀 plus 𝐁 plus 𝐂.
As a last note on the fact that vector addition is commutative, the magenta arrows show that 𝐁 plus 𝐂 plus 𝐀 is also equal to 𝐐.
