Video Transcript
Which of the vectors 𝐏, 𝐐, 𝐑, 𝐒, or 𝐓 shown in the diagram is equal to 𝚨 plus 𝚩?
The diagram referred to in the question is this set of Cartesian axes with several vectors all represented as arrows. We are trying to identify which of these arrows represents the vector that is the sum of 𝚨 and 𝚩, where 𝚨 and 𝚩 are represented in the diagram by this arrow here and this arrow here. To answer this question, we’ll need to recall how to represent a vector sum using arrows. An arrow representing a vector has a head, which is the pointy end, and a tail, which is the other end. And connecting the head and the tail is a straight line.
Now, when we want to find the sum of two vectors, we need to redraw one of those vectors so that its tail is at the head of the other vector. When we draw this picture out, we see there is one remaining tail and one remaining head. And if we connect this tail and head with a straight line, we will have another vector. This new vector is exactly the sum of the two original vectors that we were looking for. Also, we can see from this diagram that vector addition is commutative. If we draw 𝐔 with its tail at the head of 𝐕 and then connect the remaining tail and head, we get exactly the same vector as we had before. So 𝐔 plus 𝐕 is equal to 𝐕 plus 𝐔.
So as we can see from this diagram, to find 𝚨 plus 𝚩, we just need to redraw 𝚨 with its tail at the head of 𝚩 or 𝚩 with its tail at the head of 𝚨 and see which of the other vectors connects the remaining tail to the remaining head. The vector 𝚨 extends two units to the right and two units upward. So when we draw it at the head of the vector 𝚩, we get this arrow here. And the vector 𝚩 extends five units to the right and three units downward. So when we draw it at the head of the vector 𝚨, we get this arrow here. These are the two ways to add 𝚨 and 𝚩. Whichever way we draw it, the remaining tail is at the origin and the remaining head is out here at this point. And the vector 𝐒 has its tail at the origin and its head at this point. So 𝐒 is the vector in our diagram that is equal to 𝚨 plus 𝚩.
