Question Video: Determining If a Statement about Vectors Is True from a Diagram Mathematics

Video Transcript

True or False: In the given figure, vector 𝐂 is equal to vector 𝐀 plus vector 𝐁.

In this question, we’re given a figure involving three vectors, 𝐀, 𝐁, and 𝐂. And we need to determine if vector 𝐂 is equal to the sum of vectors 𝐀 and 𝐁. To do this, let’s recall what we mean by the sum of two vectors. First, a vector is an object defined entirely in terms of its magnitude and direction, and we can often think of this in terms of displacements. And we can represent this graphically by using arrows. The length of the arrow tells us the magnitude of the vector, and the direction the arrow points tells us the direction of the vector. We can then think of the addition of two vectors as combining their displacements. We just need to add the horizontal and vertical displacements together. And there’s many different ways of doing this. However, since we’re given a figure, we’re going to go through one of the methods we can use by using a figure.

Let’s start by looking entirely at vector 𝐂 in our diagram. We can find the initial point of vector 𝐂 by looking at the tail of the arrow, and we can find the terminal point of vector 𝐂 by looking at the tip of the arrow. And we can think about this in terms of displacements from the initial point of vector 𝐂 to its terminal point. We can do the exact same thing with vectors 𝐀 and 𝐁. And if we do this, we’ll notice something interesting. Let’s do the exact same thing for vector 𝐀. The initial point of vector 𝐀 is given by the tail of its arrow, and the terminal point of vector 𝐀 is given by the tip of its arrow. Similarly, for vector 𝐁, the initial point of vector 𝐁 is given by the tail of its arrow, and the terminal point of vector 𝐁 is given by the tip of its arrow.

And we can then notice something interesting. If we start at the initial point of vector 𝐀 and follow it all the way to its terminal point, we see that this is the initial point of vector 𝐁. We can then follow the vector 𝐁 all the way to its terminal point to note that this is the terminal point of vector 𝐂. In other words, if we combine the displacements of vectors 𝐀 and 𝐁, we get the exact same displacements as vector 𝐂. This is often called the tip-to-tail method of adding two vectors together, since we draw the tip of vector 𝐀 to be the same as the tail of vector 𝐁. And this allows us to conclude that the answer is true. If we add the displacements of vectors 𝐀 and 𝐁 together, we get vector 𝐂. 𝐂 is equal to 𝐀 plus 𝐁.