Video Transcript
Which of the vectors 𝐏, 𝐐, 𝐑, 𝐒, and 𝐓 shown in the diagram is equal to 𝚨 plus 𝚩?
The diagram is this set of Cartesian axes showing several vectors represented as arrows. We are interested in the sum of two of those vectors, 𝚨 and 𝚩, which are represented on the diagram by the arrows labeled here and here. And in particular, we need to identify which of the other five vectors in the diagram is equal to that sum. Now an arrow representing a vector has two ends, the pointy end, which we call the head, and the not-pointy end, which we call the tail. To add another vector to this vector, we draw the second vector with its tail at the head of this first vector. Drawn like this with the tail of one vector at the head of the other vector, we see that these two vectors almost form a triangle. They are just missing a third side.
When we draw in this third side of the triangle, we can identify the end that touches the first vector as the tail and the end that touches the second vector as the head. So this third side is also a vector. In fact, this new vector is exactly the vector that is the sum of the other two vectors. So to find the sum of two vectors using arrows, we just need to draw the tail of one of the vectors at the head of the other vector and then complete the triangle. Since we are looking for 𝚨 plus 𝚩, let’s draw 𝚩 on the diagram with its tail at the head of 𝚨.
Looking at the diagram, we see that 𝚩 extends one unit to the left and two units downward. Starting at the head of 𝚨 and drawing one unit to the left and two units downward, we see that the head of this new arrow coincides with the head of the vector 𝐑. We can also see that the tail of 𝚨 and the tail of 𝐑 are both at the origin. So 𝐑 completes the triangle formed by 𝚨 and 𝚩 exactly as we would expect from our diagram for vector addition. This tells us that the correct answer is 𝐑. 𝐑 is equal to 𝚨 plus 𝚩.
Now, vector addition is commutative, which means that the order of the vectors doesn’t affect the final answer. So if we add 𝚨 to 𝚩 by drawing this arrow here, we see that we again come to the head of the vector 𝐑, and 𝚨, 𝚩, and 𝐑 again form the appropriate triangle for vector addition. So we have verified that 𝚨 plus 𝚩 is equal to 𝚩 plus 𝚨.
