### Video Transcript

Which of the vectors π, π, π, π, and π shown in the diagram is equal to π¨ plus π©?

The diagram we are given is this set of Cartesian axes with several vectors all represented by arrows. Weβre asked in particular to consider the vectors π¨ and π© and to identify which of the other vectors is equal to the sum of π¨ and π©. So letβs recall how to add two vectors when they are represented by arrows.

An arrow representing a vector is a straight line that connects the pointy end, which we call the head, to the other end, which we call the tail. When we want to add two vectors represented this way, we draw the two of them so that the tail of one of the vectors is at the head of the other vector. These two arrows almost form a triangle. We can complete this triangle by drawing a straight line between the tail and head that are not already connected. We can make this straight line into an arrow representing a new vector if we identify the end touching the head of one of the original two vectors as the head and the other end as the tail. And this new vector is exactly the sum of the two vectors we had previously drawn.

So to find which of the vectors is equal to π¨ plus π©, we simply need to redraw π© at the head of π¨ and see which of the other five vectors completes the triangle. Now π© extends five units to the right and one unit downward. So if we draw an arrow starting at the head of π¨ and extending five units to the right and one unit downward, we see that the head of this new arrow is at the head of the vector π. Moreover, the tail of π¨ and the tail of π are both at the origin. So π correctly completes the triangle as we would expect for vector addition. This means that π is equal to π¨ plus π©.

Now vector addition is actually commutative, which means we can do it in any order. We can add π¨ plus π© or π© plus π¨ and weβll get the same answer. Letβs draw this out on our diagram. The vector π¨ extends four units to the left and four units upward. Drawing the same arrow with its tail at the head of vector π©, we extend four units left and four units upward. And the head again coincides with the head of vector π. Since the tail of π© and the tail of π are both at the origin, this drawing shows us that π is equal to π© plus π¨. So this tells us that π© plus π¨ is equal to π¨ plus π©, which confirms that vector addition is commutative.