Which of the vectors 𝐏, 𝐐, 𝐑, 𝐒, or 𝐓 shown in the diagram is equal to 𝐀 plus 𝐁?
The diagram we are given is this set of Cartesian axes, with several vectors all represented by arrows. And the question asks us to identify which of these vectors is equal to the sum of the vector 𝐀 and the vector 𝐁 shown here and here. To find the sum of 𝐀 and 𝐁, we’ll need to recall how to perform vector addition graphically.
First, let’s recall that a vector represented as an arrow has two parts. The pointy end is called the head, and the straight end is called the tail. Now we want to draw the sum of two vectors 𝐔 and 𝐕. 𝐔 and 𝐕 are both represented here as arrows, so they both have tails and they both have heads. And the way to add up these vectors is to align the tail of one vector to the head of the other vector. To align the tail of 𝐕 to the head of 𝐔, we first draw the vector 𝐔 normally, and then starting at the head of vector 𝐔, we draw the vector 𝐕. And this is aligning the tail of 𝐕 to the head of 𝐔.
Note that it would be just as valid to align the tail of 𝐔 to the head of 𝐕, and we would get the same answer because vector addition is commutative. Anyway, a vector represented by an arrow is a straight line connecting the tail and the head. And note that in the picture we’ve just drawn, we have one free tail and one free head. When we draw the straight line connecting the free tail to the free head, the vector that we draw is exactly the vector 𝐔 plus 𝐕. And this is how we add vectors graphically.
Alright, so to find 𝐀 plus 𝐁, we just need to redraw the vector 𝐀 with its tail at the head of the vector 𝐁 or, vice versa, redraw the vector 𝐁 with its tail at the head of the vector 𝐀. Looking at the diagram, the vector 𝐀 extends from the origin one unit to the left and two units upward. So we just need to draw an arrow starting at the head of the vector 𝐁 that extends one unit to the left and two units upward. Here is that arrow, and we can see that its head is in the same location as the head of the vector 𝐓. And all of the vectors’ tails meet at the origin, so 𝐓 has its tail at the tail of 𝐁 and its head at the head of 𝐀. And so 𝐓 must be the vector 𝐀 plus 𝐁.
Let’s double-check our answer by drawing the vector 𝐁, starting at the head of the vector 𝐀. The arrow corresponding to the vector 𝐁 looks like this. And as we can see, its head is at the head of the vector 𝐓, its tail is at the head of the vector 𝐀, and the tail of the vector 𝐀 is at the tail of the vector 𝐓. So, again, 𝐀 plus 𝐁 is equal to 𝐓.