Question Video: Measuring the Magnitude of a Resultant Vector Physics

Video Transcript

Some vectors are drawn to the scale of the ruler on a square grid. The sides of the squares are one centimeter long. The red vector is the resultant of the blue and green vectors. What is the length of the resultant vector, measured to the nearest centimeter?

Okay, so in this question, we’re given a diagram that has three vectors in, and we’re told that the red vector is the resultant of the blue and green vectors. We’re also told that the vectors are drawn to a scale and that the sides of the squares in the diagram are one centimeter long. We are then asked to find the length of the resultant vector. Let’s begin by recalling that the resultant of two vectors is the vector that is found by adding them together and that two vectors may be added by drawing them tip to tail. Recall that the tail of a vector is where it starts and the tip of a vector is where it extends or points to. So drawing two vectors tip to tail means drawing the second vector with its tail starting at the tip of the first vector like this.

Then we can find the sum of these two vectors or the resultant by drawing an arrow from the tail of the first vector to the tip of the second vector. So in this example, the blue arrow that we have just added to the diagram is our resultant vector.

Okay, now that we’ve seen what is meant by a resultant vector, let’s look back to the question. We see that we are told that the red vector is the resultant of the blue and green vectors. If we look at the diagram, we see that the green vector is drawn with its tail at the tip of the blue vector. So the blue and green vectors are drawn tip to tail. If we now look at the red vector, we see that it has its tail at the tail of the first vector, the blue vector, and its tip at the tip of the second vector, the green vector. And so we see that this red vector is indeed the resultant of the blue and green vectors.

In this question, we see that we have a blue vector that is entirely horizontal and a green vector that is entirely vertical. This means that the angle between these two vectors is 90 degrees, so we can see that our three vectors in the diagram form a right-angled triangle. The question is asking us to find the length of this resultant vector, which means finding the length of the hypotenuse of the right-angled triangle. In order to do this, let’s recall Pythagoras’s theorem. If we label the lengths of the sides of the triangle 𝑎, 𝑏, and 𝑐, where 𝑐 is the hypotenuse, then Pythagoras’s theorem tells us that 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared.

Since in this question we’re trying to find the value of 𝑐, let’s make 𝑐 the subject by taking the square root of both sides of this equation. We then have that 𝑐 is equal to the square root of 𝑎 squared plus 𝑏 squared. What this equation is telling us is that if we want to find the value of 𝑐, the length of the hypotenuse of the triangle, then we need to know the values of 𝑎 and 𝑏, the lengths of the other two sides. Luckily for us, we have a scale on our diagram, and the vectors 𝑎 and 𝑏 both point along the lines of the grid, which makes it easy to read off their lengths. We are told that the squares in the diagram have sides that are one centimeter long. In the diagram itself, we have a ruler showing these one-centimeter marks in a vertical direction.

And of course, since we’re told that the grid consists of squares and if they are one centimeter in the vertical direction, they must also be one centimeter in the horizontal direction. So one square’s worth of distance in either the horizontal or the vertical direction corresponds to one centimeter. This means, for both the blue vector and the green vector, that in order to find the length of the vector, we simply need to start at the tail and count the number of squares until we reach the tip. This number of squares then gives the length of that vector measured in centimeters.

Let’s begin with the blue vector. We start at the tail of the vector and we count the number of squares until we reach the tip of the vector. In this case, we find that that number of squares is five. So we can say that 𝑎, the length of this blue vector, is equal to five centimeters. Now let’s look at the green vector. We’ll start at the tail of this vector, which is placed at the tip of the blue vector, and we’ll count the number of squares until we reach the tip of this vector. And in this case, we find that that number of squares is 11. So we can say that 𝑏 is equal to 11 centimeters. Now that we have our values of 𝑎 and 𝑏, we simply need to substitute them into this equation in order to calculate 𝑐.

If we substitute in that 𝑎 equals five centimeters and 𝑏 equals 11 centimeters, then we get that 𝑐 is equal to the square root of five centimeters squared plus 11 centimeters squared. When doing this calculation, we should take care with our units because if we take the square of a quantity with units of centimeters, we’re going to get a quantity with units of centimeters squared. So in this case, if we take the square of five centimeters, we get 25 centimeters squared. And if we take the square of 11 centimeters, we get 121 centimeters squared. If we then add together 25 centimeters squared and 121 centimeters squared, we get that 𝑐 is equal to the square root of 146 centimeters squared.

Then the last thing left to do is to evaluate the square root. If we take the square root of a quantity with units of centimeters squared, then we get a result with units of centimeters. And since we know that 𝑐 is meant to be a length, the length of this red vector in our diagram, it makes sense that it should have units of distance. And if we take the square root of 146, we get a result of 12.083 and so on with more decimal places. And this result that we have found gives us the length of our resultant vector. But if we look back at the question, we see that we are asked to give the length to the nearest centimeter. Rounding 12.083 to the nearest centimeter gives a result of 12 centimeters.

And so we have our answer to the question that the length of the resultant vector, measured to the nearest centimeter, is equal to 12 centimeters.