Some vectors are drawn to the scale of the ruler in the diagram. The squares of the grid have sides one centimeter in length. 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?
Right. So in this question, we’re given a scale diagram that has three vectors in. And we’re told that the red vector is the resultant of the blue and green vectors. We are also told that the grid squares have sides that are one centimeter in length, and we’re asked to find the length of the resultant vector. Let’s recall that the resultant of two vectors is the vector that is found by adding those two vectors together and that we can add two vectors by drawing them tip to tail.
Remember 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.
Now that we’ve seen what’s meant by a resultant vector, let’s get back to the question. The question tells us that the red vector is the resultant of the blue and green vectors. If we look at the diagram, we should see that the blue and green vectors are drawn tip to tail. Notice that the green vector is drawn starting with its tail at the tip of the blue vector. If we now look at the red vector, we see that it’s drawn with its tail at the tail of the first vector, the blue vector, and that its tip is at the tip of the second vector, the green vector. So we see that this red vector is indeed the resultant of the blue and green vectors.
We can see in this case that our blue vector is entirely horizontal and our green vector is entirely vertical. This means that we know that the angle between these two vectors is equal to 90 degrees. And so we can see that the three vectors 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 find this hypotenuse, let’s recall Pythagoras’s theorem. If we label the lengths of the sides of this 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 take the square root of both sides of this equation to get an equation where 𝑐 is the subject. We then have that 𝑐 is equal to the square root of 𝑎 squared plus 𝑏 squared.
This equation is telling us that if we want to find the value of 𝑐, then we need to know the values of 𝑎 and 𝑏. Remember that 𝑎 is the length of the blue vector in our diagram and 𝑏 is the length of the green vector. 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 of the grid have sides that are one centimeter in length. In the diagram itself, we have a ruler showing these one-centimeter increments in the vertical direction. And of course, since we’re told that the grid consists of squares, then if these squares 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 a one centimeter. What this means for us is that in order to find the lengths of each of the blue vector and the green vector, all we need to do is start at the tail of the vector and count the number of squares until we reach the tip of the vector. 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 that vector. And in this case, we find that that number of squares is equal to 10. So this means that the length of the blue vector and the value of 𝑎 is equal to 10 centimeters.
Now let’s look at the green vector. We start at the tail of this vector, which is placed at the tip of the blue vector. And we count the number of squares until we reach the tip. We find that this number of squares is also 10. And so the length of our green vector and the value of 𝑏 is also equal to 10 centimeters. We now have values for both 𝑎 and 𝑏, so we can substitute these values into our equation for 𝑐.
If we substitute in that 𝑎 equals 10 centimeters and 𝑏 equals 10 centimeters, then we get that 𝑐 is equal to the square root of 10 centimeters squared plus 10 centimeters squared. When doing this calculation, we need to take some care with our units because if we take the square of a quantity with units of centimeters, then we’re going to get a quantity with units of centimeters squared. In this case, taking the square of 10 centimeters gives us 100 centimeters squared. And if we add together 100 centimeters squared and 100 centimeters squared, we get 200 centimeters squared.
The last step left to go is to evaluate the square root. If we take the square root of a quantity with units of centimeters squared, we get a result with units of centimeters. As a quick common-sense check, we know that 𝑐 is meant to be a length; it’s the length of this red vector in our diagram. So having units of centimeters does make sense. And if we take the square root of 200, we get a result of 14.142 and so on with further decimal places.
And this result we found here for the value of 𝑐 gives us the length of the resultant vector, which is what they were asking us for in the question. But if we look back to the question, we see that we were asked to give our result to the nearest centimeter. Rounding 14.142 to the nearest centimeter gives us 14 centimeters. And so our answer to the question is that the length of the resultant vector measured to the nearest centimeter is 14 centimeters.