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Question Video: Comparing the Magnitudes of Parallel and Perpendicular Vectors Physics

Some vectors that represent forces are drawn to scale on a square grid. The sides of the squares are one centimeter long. A distance of one centimeter on the grid represents one newton of force. What is the difference between the magnitude of the larger horizontal force and the magnitude of the smaller horizontal force? What is the difference between the magnitude of the larger vertical force and the magnitude of the smaller horizontal force?

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Video Transcript

Some vectors that represent forces are drawn to scale on a square grid. The sides of the squares are one centimeter long. A distance of one centimeter on the grid represents one newton of force. What is the difference between the magnitude of the larger horizontal force and the magnitude of the smaller horizontal force? What is the difference between the magnitude of the larger vertical force and the magnitude of the smaller horizontal force?

Okay, so in this question, we’re given a diagram in which we have four different vectors that are drawn to scale on a square grid, and we’re told that these vectors represent forces. We’re told that the sides of the squares of this grid are one centimeter long and that one centimeter on the grid represents one newton of force. There are then two parts to the question, both of which are asking us to calculate a difference between magnitudes of two of the forces represented by vectors shown in the diagram. The first part of the question is asking us about the larger horizontal force and the smaller horizontal force. Then the second part of the question is asking about the larger vertical force and the smaller horizontal force. So let’s see if we can identify all of these in the diagram.

We’ll begin by recalling that a vector is a quantity that has both a magnitude and a direction. And when we represent a vector by an arrow, the direction of the vector is the direction of the arrow and the magnitude of the vector is the length of the arrow. If we look at our diagram, we see that we have arrows pointing in all kinds of different directions. We have a green and a purple arrow pointing upwards, a red arrow pointing to the left, and a blue arrow pointing to the right. So we can identify the green and the purple arrows as being vertical. And we can identify the red and the blue arrows as being horizontal.

To identify which is the larger horizontal force and which is the smaller horizontal force, we’re going to need to work out the magnitudes of those forces because the larger horizontal force is the horizontal force with the larger magnitude. We’re told that we have a square grid and that all of our forces are drawn to scale. If we look at the diagram, we can see that all of the force vectors are drawn along the lines of the grid, which makes it easy to work out their lengths. We are told that the side of each square is one centimeter long and that one centimeter represents one newton. So one square of length corresponds to one newton of force.

Or, in other words, to get the magnitude of any of the force vectors shown in our diagram, we simply need to count how many squares long that vector is, and that number gives the magnitude of the force measured in newtons. So let’s work out the magnitudes of the forces shown in the diagram. We’ll begin with the red vector. If we start at one end of this vector and count the number of squares until we reach the other end, we find that this vector is 12 squares in length. And so the force represented by this vector has a magnitude of 12 newtons.

Now let’s look at the purple vector. This vector is going vertically. Again, let’s start at one end of the vector and count the number of squares until we get to the other end. In this case, that number of squares is five, and so the purple vector represents a force of magnitude five newtons. If we apply the same process to the blue vector and count the number of squares between one end of the vector and the other, we find that this one is 10 squares in length and so represents a force of magnitude 10 newtons. Finally, for the green vector, we find that it is eight squares in length and so represents a force with a magnitude of eight newtons.

Okay, let’s summarize what we’ve found. We have two horizontal forces: the red arrow with a magnitude of 12 newtons and the blue arrow with a magnitude of 10 newtons. Since 12 newtons is bigger than 10 newtons, we can identify the red arrow as being the larger horizontal force and the blue arrow as representing the smaller horizontal force. Similarly, we have two vertical forces, the green arrow with a magnitude of eight newtons and the purple with a magnitude of five newtons. And since eight newtons is greater than five newtons, we can identify the green arrow as representing the larger vertical force and the purple arrow as representing the smaller vertical force.

In the first part of the question, we’re asked for the difference between the magnitude of the larger horizontal force and the smaller horizontal force. And we are now in a position to identify which those two forces are. We see that we have two horizontal forces and that we have identified that the force represented by the red arrow has a larger magnitude than the force represented by the blue arrow.

We are asked to find the difference between the magnitude of the larger horizontal force and the smaller horizontal force. To calculate a difference in magnitudes, we need to subtract the smaller magnitude from the larger. And so here we have that the magnitude of the larger horizontal force is equal to 12 newtons. And we need to subtract from that the magnitude of the smaller horizontal force, which is 10 newtons. And if we do this subtraction, we get a result of two newtons. This is our answer to the first part of the question.

The second part of the question is asking for the difference between the magnitude of the larger vertical force and the smaller horizontal force. Now, we’ve already identified that the blue arrow represents the smaller horizontal force. So now we just need to identify which is the larger vertical force. We see that we have two vertical forces and that the green arrow represents a larger magnitude force than the purple arrow. So this time we’re calculating the difference in magnitude of the force represented by the blue arrow, which has a magnitude of 10 newtons, and the force represented by the green arrow, which has a magnitude of eight newtons.

In order to find the difference in magnitudes between these two forces, we take the larger of the magnitudes, which is 10 newtons, and we subtract from this the smaller magnitude, which is eight newtons. If we do this subtraction, we get a result of two newtons. And so our answer to the second part of the question is also two newtons.

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