Question Video: Finding the Resultant of Vectors on a Scale Diagram Physics • First Year of Secondary School

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. The difference between the magnitude of the larger horizontal force and the magnitude of the smaller horizontal force is Δ𝐅 one. The difference between the magnitude of the larger vertical force and the magnitude of the smaller horizontal force is Δ𝐅 two. What is the absolute value of Δ𝐅 one minus the absolute value of Δ𝐅 two?

The question asks us about the following quantity: the absolute value of Δ𝐅 one minus the absolute value of Δ𝐅 two. But what does the absolute value of Δ𝐅 one minus the absolute value of Δ𝐅 two represent? Let’s note that Δ𝐅 one and Δ𝐅 two are both differences of magnitudes of vectors. This means that Δ𝐅 one and Δ𝐅 two are scalars; they are not vectors. So, the absolute value of Δ𝐅 one minus the absolute value of Δ𝐅 two is the difference of the absolute value of two scalars, not the result of adding two vectors.

That said, let’s continue with our problem. The question tells us that the side of a square measures one centimeter and that the distance of one centimeter represents a newton of force. Now we count the squares that measure each vector. The red force vector has a length of 12 spaces and therefore of 12 centimeters. It then represents a magnitude of 12 newtons. We will say that 𝐅 sub 𝑅 equals 12 newtons, which is the magnitude of the red vector. The vector is horizontal and points to the left.

The other horizontal vector is the blue vector. This vector points to the right. Now we count how many squares its length is. There are 10 squares, meaning 10 centimeters. Therefore, the magnitude of the blue vector is 10 newtons. So, 𝐅 sub 𝐵 equals 10 newtons. Now we have the value of Δ𝐅 one, which is the difference between the magnitude of the horizontal longest force vector and the magnitude of the horizontal shortest vector. We see then that Δ𝐅 one equals 12 newtons minus 10 newtons, which equals two newtons.

The question also asks for the difference between the magnitude of the larger vertical force and the magnitude of the smaller horizontal force. If we now consider vertical vectors, we can see that for the longest vertical vector of green color, 𝐅 sub 𝐺 equals eight newtons because it measures eight centimeters. And for the shortest vertical vector of purple color, 𝐅 sub 𝑃 equals five newtons because it measures five centimeters.

The magnitude of the longest vertical vector is required. This magnitude is compared to the magnitude of the smaller horizontal vector. We have already seen that the smaller horizontal vector is 𝐅 sub 𝐵, which has a magnitude of 10 newtons. So, Δ𝐅 two equals 10 newtons minus eight newtons, which equals two newtons.

We can now calculate the absolute value of Δ𝐅 one minus the absolute value of Δ𝐅 two. This will equal the absolute value of two newtons minus the absolute value of two newtons, which equals zero newtons. And so, the absolute value of Δ𝐅 one minus the absolute value of Δ𝐅 two is zero newtons.