Video Transcript
The diagram shows three vectors 𝐀, 𝐁, and 𝐂. Which of the following expressions gives 𝐂 in terms of 𝐀 and 𝐁? Is it (a) 𝐂 is equal to two 𝐁 minus 𝐀, (b) 𝐂 is equal to 𝐀 plus 𝐁, (c) 𝐂 is equal to two 𝐀 minus 𝐁, (d) 𝐂 is equal to 𝐁 minus 𝐀, or (e) 𝐂 is equal to 𝐀 minus 𝐁?
If we look at our diagram, we can see that both 𝐀 and 𝐁 start at the origin and 𝐂 goes from the tip of vector 𝐀 to the tip of vector 𝐁. Now recall that when we add two vectors, we place them end to end so that the tail of the second vector meets the tip of the first one. The sum of those two vectors is then the vector that goes from the tail of the first vector to the tip of the second.
In this example, we can see that the tip of vector 𝐀 meets the tail of vector 𝐂. And then vector 𝐁 goes from the tail of vector 𝐀 to the tip of vector 𝐂. We can therefore say that the sum of vector 𝐀 plus vector 𝐂 is equal to vector 𝐁. We can then rearrange this in terms of 𝐂 by subtracting 𝐀 from both sides. And we find that 𝐂 is equal to 𝐁 minus 𝐀. Therefore, the correct answer is (d) 𝐂 is equal to 𝐁 minus 𝐀.
Another way of looking at this is we could start from the tail of vector 𝐂. And we can reach the tip by going backwards along 𝐀 and then forwards along 𝐁. In other words, we take the negative of vector 𝐀 plus vector 𝐁 to make vector 𝐂. And then we can just swap these two around. And this gives us another method of finding the same answer that 𝐂 is equal to 𝐁 minus 𝐀.
