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 statement is this set of Cartesian axes here with several vectors all represented as arrows. The question asks us to identify which of the arrows in the diagram represents the vector that is equal to 𝚨 plus 𝚩. And 𝚨 plus 𝚩 are also represented in the diagram with this arrow here and this arrow here. To answer this question, we will need to understand how to add two vectors graphically. First, recall that when we represent vectors graphically, they have two parts: the tail and the head.
Now, say we have two vectors called 𝐔 and 𝐕. To find the sum of 𝐔 and 𝐕, we align the tail of one of the vectors to the head of the other vector. So, if we’re trying to align the tail of 𝐕 to the head of 𝐔, we simply redraw the vector 𝐕 but with its tail at the head of 𝐔. And here’s what that looks like. This arrow is exactly the same as the arrow that represents the vector 𝐕 except that its tail is at the head of the arrow that represents the vector 𝐔. We’re now ready to draw the arrow that represents the vector 𝐕 plus 𝐔. This vector has its tail at the tail of 𝐔 and its head at the head of 𝐕.
Now, recall that vector addition is commutative, so 𝐔 plus 𝐕 is equal to 𝐕 plus 𝐔. And we can see this in our diagram. If we draw the vector 𝐔 with its tail at the head of the vector 𝐕, we see that the vector we have already drawn has its tail at the tail of 𝐕 and its head at the head of 𝐔. So it must be the vector 𝐕 plus 𝐔. This diagram shows us graphically that vector addition is commutative.
Anyway, the question asks us to find the arrow that represents the vector 𝚨 plus 𝚩. So all we need to do is draw a copy of the arrow representing the vector 𝚩 with its tail at the head of the arrow representing the vector 𝚨. The arrow representing 𝚩 extends five units to the left and one unit upward. Drawing this same arrow with its tail at the head of the arrow representing the vector 𝚨, we see that our arrow points directly to the head of the vector 𝐑. Now, all of the vectors in the diagram have their tails at the same point, the origin of the Cartesian axes. So the tail of 𝐑 is at the tail of 𝚨. And as we’ve already seen, the head of 𝐑 is at the head of 𝚩. So 𝐑 must be the vector 𝚨 plus 𝚩.
We can confirm this answer by using the fact that 𝚨 plus 𝚩 is equal to 𝚩 plus 𝚨. The vector 𝚨 extends two units to the right and four units upward. So, if we draw an arrow with its tail at the head of the vector 𝚩 and extending two units to the right and four units upward, we also come to the head of the vector 𝐑. And again, since 𝐑 and 𝚩 share the same tail and 𝐑 and this new arrow representing 𝚨 share the same head, 𝐑 is equal to 𝚨 plus 𝚩.
