Fill in the blank: In triangle 𝐴𝐵𝐶, the vector 𝐀𝐁 plus the vector 𝐁𝐂 is equal to what?
In this question, we’re looking at some geometric properties of vectors. So what we’re going to do is begin by sketching out triangle 𝐴𝐵𝐶. Let’s suppose triangle 𝐴𝐵𝐶 look like this. The vector 𝐀𝐁 takes us from point 𝐴 to point 𝐵 in a straight line. And so the vector 𝐀𝐁 is the line denoted with the yellow arrow. Similarly, the vector 𝐁𝐂 takes us from point 𝐵 to point 𝐶 in a straight line. So we denote the vector 𝐁𝐂 with the pink arrow. Then the sum of these two vectors is said to be its resultant. If we think about each vector as a journey, the vector 𝐀𝐁 takes us from 𝐴 to 𝐵 and the vector 𝐁𝐂 takes us from 𝐵 to 𝐶, the resultant is the complete journey that we take.
So let’s identify what’s happened. We start at point 𝐴, we then move to point 𝐵 along vector 𝐀𝐁, and then we move along vector 𝐁𝐂 to point 𝐶. The resultant can be thought about as the movement that takes us from 𝐴 to 𝐶. This time, that’s the green arrow. Since we have a single straight line that takes us from 𝐴 to 𝐶, we can say that the resultant of 𝐀𝐁 and 𝐁𝐂 is the vector 𝐀𝐂. And so in triangle 𝐴𝐵𝐶, the vector 𝐀𝐁 plus the vector 𝐁𝐂 is the vector 𝐀𝐂.