### Video Transcript

Which of the following correctly shows a vector represented by a dashed arrow that is the resultant of two vectors shown by solid arrows?

The question shows us four different diagrams. In each of these diagrams, there are two vectors represented by solid arrows, one of which is blue and the other is red. These solid arrows are the same in each of the four diagrams. And in each case, the two vectors are drawn tip to tail. That means that one vector is drawn starting with its tail at the position of the tip of the first vector. So, for example, looking at diagram (A), we can see that the blue vector starts from its tail over here and extends to its tip over here and that the red vector is drawn starting with its tail at the position of the tip of the blue vector.

We’re told that in these diagrams, the dashed arrow is supposed to represent the vector that is the resultant of the two vectors shown by the solid arrows. We can recall that whenever we have two vectors represented by arrows that are drawn tip to tail, like the blue and the red arrow in our diagrams, then the resultant of these two vectors is represented by an arrow that has its tail at the position of the tail of the first vector and its tip at the position of the tip of the second vector. So, that’s an arrow that points from the tail of the first vector to the tip of the second vector.

Looking at diagram (A), we can see that the dashed arrow starts with its tail at the tail of the blue vector but that its tip is over here, which is not at the tip of the red vector. So, the dashed arrow in diagram (A) does not point from the tail of the blue arrow to the tip of the red arrow, which means that it can’t be the resultant of these two vectors. Then, we know that diagram (A) can’t be the correct answer.

If we look now at diagram (B), we can see that the black dashed arrow starts with its tail over here at the position where the tip of the blue arrow meets the tail of the red arrow. This dashed arrow then extends so that its tip is over here. Clearly, this black dashed arrow is not directed from the tail of the first arrow — that’s the blue one — to the tip of the second arrow, the red one. So, it cannot represent the resultant of the two vectors, which means that we can rule out diagram (B).

Now, let’s consider the diagram shown in option (C). In this case, the tail of the black dashed arrow is here at the tip of the red arrow. And the tip of this dashed arrow is at the tail of the blue arrow over here. So, the black dashed arrow is pointing from the tip of the second red arrow to the tail of the first blue arrow. In other words, it’s connecting the correct two positions, but it’s doing so in the wrong direction. This means that diagram (C) cannot be correct. And in fact, in this case, the black dashed arrow represents the negative of the resultant of the vectors represented by the red and the blue arrows.

This leaves us with the diagram shown in option (D). In this case, we can see that the black dashed arrow starts with its tail over here at the tail of the blue arrow and extends so that its tip is over here at the tip of the red arrow. So, this dashed arrow is pointing from the tail of the first arrow to the tip of the second one, which means that it represents the resultant of the two vectors shown by these solid arrows. So, our answer is that the correct diagram is this one, which is shown in option (D).