### Video Transcript

Which of the following is equivalent to one-half times the vector from ๐ถ to ๐ต minus the vector from ๐ท to ๐ถ? Is it option (A) the vector from ๐ด to ๐ธ? Option (B) the vector from ๐ต to ๐ธ. Option (C) the vector from ๐ถ to ๐ธ. Option (D) the vector from ๐ถ to ๐ด. Or is it option (E) the vector from ๐ธ to ๐ถ?

In this question, we need to determine which of five given vectors is equivalent to a given expression involving vectors. And we want to do this by using the given diagram. Letโs start by looking at the given vector expression weโre given. This vector expression is in two parts, first when multiplying an entire expression by one half, and inside of our parentheses, we can see we have the difference between two vectors. Letโs start by looking at the inner parentheses since this is the first operation carried out, vector from ๐ถ to ๐ต minus vector from ๐ท to ๐ถ.

To find the difference of two vectors, we can do this by using the diagram. However, letโs start by sketching these two vectors on our given diagram. Letโs start with the vector from ๐ถ to ๐ต. Its initial point will be point ๐ถ and its terminal point will be vector ๐ต, so we draw an arrow from ๐ถ to ๐ต as shown. And itโs worth recalling here vectors are defined entirely in terms of their magnitude and direction. So only the direction of this arrow and the length of this arrow define the vector.

So, for example, by using our knowledge of parallelograms, we could draw the vector from ๐ถ to ๐ต starting at point ๐ท and ending at point ๐ด. We know that these two vectors have the same direction since ๐ด๐ต๐ถ๐ท is a parallelogram, so the side ๐ด๐ท is parallel to the side ๐ต๐ถ. Similarly, we know they have the same magnitude since opposite sides in a parallelogram have the same length. Letโs follow the same process to sketch the vector from ๐ท to ๐ถ. The initial point of this vector is vertex ๐ท and the terminal point of this vector is vertex ๐ถ. This gives us the following vector from ๐ท to ๐ถ. We could follow the same reasoning to determine the vector from ๐ท to ๐ถ is equivalent to the vector from ๐ด to ๐ต.

However, before we do this, letโs look at the expression we need to evaluate. Inside our parentheses, weโre subtracting the vector from ๐ท to ๐ถ from the vector from ๐ถ to ๐ต. And thereโs several different ways we can evaluate this expression. However, the easiest way is to notice that weโre subtracting one vector from another, and itโs usually easier to add vectors together. And we can recall multiplying a vector by negative one switches its direction. However, it leaves its magnitude unchanged. So in this case, negative one times the vector from ๐ท to ๐ถ will be equal to the vector from ๐ถ to ๐ท. And of course, this result holds true in general. Negative one times a vector from a point ๐ to a point ๐ will just be the vector from ๐ to ๐. Therefore, we have that the vector from ๐ถ to ๐ต minus the vector from ๐ท to ๐ถ is equal to the vector from ๐ถ to ๐ต added to the vector from ๐ถ to ๐ท.

We can also sketch this on our diagram by switching the direction of our vector. This then gives us the following. We now want to evaluate the vector from ๐ถ to ๐ต added to the vector from ๐ถ to ๐ท. We can do this by recalling how we add two vectors together graphically. This is sometimes called the triangle rule for vector addition. If we have two vectors ๐ฎ and ๐ฏ drawn tip to tail so the terminal point of vector ๐ฎ is the initial point of vector ๐ฏ, then the vector ๐ฎ added to the vector ๐ฏ is the vector which starts at the initial point of vector ๐ฎ and finishes at the terminal point of vector ๐ฏ.

Another way of thinking about this is we can add two vectors drawn in this way by following the arrows since weโre just adding their displacements. We want to use this to add the vector from ๐ถ to ๐ต to the vector from ๐ถ to ๐ท. And we can do this by using our diagram and our knowledge of vector addition.

There are a few different ways of doing this. However, weโll only go through one of these. Letโs start by looking at the vector from ๐ถ to ๐ต. It starts at vertex ๐ถ and ends at vertex ๐ต. So to add this to the vector from ๐ถ to ๐ท by using our rule, we need the vector from ๐ถ to ๐ท to have its initial point at vector ๐. And we can do this by using one of our previous results. Opposite sides in a parallelogram have the same direction since theyโre parallel and they have the same magnitude. This means the vector from ๐ต to ๐ด must be equal to the vector from ๐ถ to ๐ท since ๐ด๐ต๐ถ๐ท is a parallelogram.

And now, we can easily add these two vectors together by noticing the terminal point of the vector from ๐ถ to ๐ต is coincident with the initial point of the vector from ๐ต to ๐ด, which is the same as the vector from ๐ถ to ๐ท. This means the vector from ๐ถ to ๐ต added to the vector from ๐ถ to ๐ท will be the vector with initial point at ๐ถ and terminal point at ๐ด. However, weโre not done yet. Remember, weโre only working with the expression inside of the parentheses. So this is only the vector ๐ถ๐ต minus the vector ๐ท๐ถ.

We still need to multiply this vector by one-half. And we can do this by recalling multiplying a vector by a positive scalar does not affect its direction; it only affects its magnitude. In particular, multiplying a vector by one-half halves its magnitude. So we need to find a vector in the same direction as this vector. However, it needs to have half the magnitude. And we can do this by noting weโre given the bisector of the line segment from ๐ถ to ๐ด. Itโs point ๐ธ. Therefore, the vector from ๐ธ to ๐ด and the vector from ๐ถ to ๐ธ will be half the magnitude of this vector in the same direction. And of these two options, we can see that only the vector from ๐ถ to ๐ธ appears in the five given options.

Therefore, we were able to show of the five given options, only option (C) the vector from ๐ถ to ๐ธ is equivalent to a half times the vector from ๐ถ to ๐ต minus the vector from ๐ท to ๐ถ.