# Question Video: Adding Three Vectors Graphically Physics

Which of the vectors 𝐏, 𝐐, 𝐑, 𝐒, and 𝐓 shown in the diagram is equal to 𝐀 + 𝐁 + 𝐂?

03:04

### Video Transcript

Which of the vectors 𝐏, 𝐐, 𝐑, 𝐒, and 𝐓 shown in the diagram is equal to 𝐀 plus 𝐁 plus 𝐂?

When we represent a vector with an arrow, we call the pointy end the head and the other end the tail. To add a second vector to this first vector, we start by drawing the second vector with its tail at the head of the first vector. If we now draw a third vector with its tail at the tail of the first vector and its head at the head of the second vector, this third vector is exactly the sum of the two vectors we originally drew.

Now, this is all well and good, but the question asks us to find the sum of three vectors, not just two. However, vector addition is associative. So adding three vectors together is the same as adding two vectors and then adding the third vector to that sum. So, to add a third vector to 𝐔 and 𝐕, we simply draw that third vector with its tail at the head of 𝐔 plus 𝐕. And then we’ll connect the tail of 𝐔 plus 𝐕 to the head of this third vector. We’ve drawn our third vector 𝐖. And now we draw our final vector representing the total sum with its tail at the tail of 𝐔 plus 𝐕 and its head at the head of 𝐖.

Now that we have the final sum, when we erase the intermediate vector 𝐔 plus 𝐕, we see that by connecting the three vectors from tail to head, a fourth vector with its tail at the remaining tail and its head at the remaining head is exactly equal to the sum of these three vectors. In fact, this works for any number of vectors. And for any particular set of vectors, it doesn’t matter what order we draw them in. We will always get the same answer because vector addition is both commutative and associative.

Returning to the diagram, we see that to answer this question, we need to connect 𝐀, 𝐁, and 𝐂 from tail to head and then find which of the other vectors connects the remaining tail to the remaining head. Since order doesn’t matter, we may as well connect the tail of 𝐁 to the head of 𝐀 and then the tail of 𝐂 to the head of 𝐁. 𝐁 extends one unit to the left and five units upward. So, starting at the head of 𝐀, we draw an arrow that extends one unit to the left and five units upward.

Now we need to include 𝐂. 𝐂 extends four units to the left and four units downward. When we connect this arrow to the arrows we’ve already drawn, we see that the head coincides with the head of the vector 𝐓. Now, this head, the head of the arrow we’ve labeled 𝐂, is our remaining head. And our remaining tail is the tail of the vector 𝐀 located at the origin. And as we can see, the vector 𝐓 has its tail at the origin and its head at our remaining head. So 𝐓 must be the sum of 𝐀, 𝐁, and 𝐂.

If we were to add the vectors in a different order, perhaps 𝐂 then 𝐀 then 𝐁 as shown, we would find that the remaining head still coincides with the head of 𝐓 and the remaining tail still coincides with the tail of 𝐓, which confirms that our answer is 𝐓 and also shows visually how vector addition is commutative.