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 resulting vector measured to the nearest centimeter?
Okay, so in this question, we’re given a diagram that has three vectors in. We’re told that the red vector is the resultant of the blue and the green vectors. We’re also told that the squares of the grid in the diagram have sides that are one centimeter in length. And we are asked to work out the length of the resultant vector.
Let’s begin by recalling that the resultant of two vectors is found by adding those two vectors together and that we can add two vectors together 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. 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 vectors or their resultants by drawing an arrow from the tail of the first vector to the tip of the second vector. So in this example, this blue arrow that we’ve just added is the resultant vector.
Looking back at the question, we see that we’re told that the red vector is the resultant of the blue and the green vectors. And if we look at our diagram, we see that we’ve got the green vector with its tail drawn starting at the tip of the blue vector. So we’ve got the blue and green vectors drawn tip to tail. And the red resultant vector is indeed drawn with its tail starting at the tail of our first vector, the blue vector, and extending to the tip of our second vector, the green vector. And in this case, our blue vector is entirely horizontal and our green vector is entirely vertical. So the angle between these two vectors is 90 degrees.
So we can see that our three vectors form a right-angled triangle. We’re asked to find the length of this resultant vector, which means that we need to find 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 given by 𝑎 squared plus 𝑏 squared.
Since in this question we’re trying to find the value of 𝑐, we can take 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. So what this equation is telling us is that in order to find 𝑐, the length of our resulting vector, we need to know the values of 𝑎 and 𝑏, the lengths of the blue vector and the green vector. Luckily for us, we have a scale on our diagram. And since the blue and green vectors point along the lines of the diagram, this makes it easy to read off their lengths.
We’re told in the question that each of the squares in the diagram has sides that are one centimeter in length. And this information is reinforced by the presence of a ruler in the diagram showing that the lines of the grid in this diagram are spaced one centimeter apart. So to find the lengths of each of 𝑎 and 𝑏, we simply need to start at the tail of the vector and count the number of squares until we reach the tip of the vector.
Let’s begin with vector 𝑎. The tail of the vector is at this point marked here. And if we count the number of squares until we reach the tip of the vector, we find that vector 𝑎 is 10 squares in length. And since we know that one square corresponds to one centimeter, we know that 𝑎 is equal to 10 centimeters. Now let’s do the same for vector 𝑏. The tail of 𝑏 is at the tip of 𝑎. And if we count the number of squares until we reach the tip of 𝑏, we find that 𝑏 is also 10 squares in length. And again, since we know that each square is one centimeter, this means that 𝑏 is also equal to 10 centimeters.
Now that we have our values for 𝑎 and 𝑏, we can substitute them in to our equation for 𝑐. Doing this, we have that 𝑐 is equal to 10 centimeters squared plus 10 centimeters squared. When doing this calculation, we need to take a little care over the units because if we take the square of 10 centimeters, then we get 100 with units of centimeters squared. If we then add together 100 centimeters squared and 100 centimeters squared, we get that 𝑐 is equal to the square root of 200 centimeters squared.
So all that’s left to do is to evaluate this 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 if we take the square root of 200, we get a value of 14.142 and so on with further decimal places. And this result gives us the length of our resulting vector. Looking back at the question, we see that we were asked to give our answer to the nearest centimeter. So our result rounds to 14 centimeters. So we have our answer to the question that, to the nearest centimeter, the length of the resultant vector in the diagram is equal to 14 centimeters.