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?
Okay, so in this question, we are given a diagram that has three vectors in. And we’re told that the red vector is the resultant of the blue and the green vectors. We are also told that the squares in the diagram have sides that are one centimeter in length. And we are asked to find the length of the resultant vector. Let’s recall that the resultant of two vectors is the vector that we get when we add those two vectors together. When we have vectors drawn on a scale diagram, we can add together those vectors by drawing them tip to tail.
We can identify the tail and the tip of a vector as follows. The tail is where that vector starts, and the tip is where that vector points or extends to. So drawing two vectors tip to tail means drawing the tail of one vector starting at the tip of the other. Then the resultant is what we get when we add these two vectors together. It starts at the tail of the first vector and extends to the tip of the second vector. So in this example, the resultant vector is this blue arrow here. And we see that the 𝑥-component of our resultant vector is given by the sum of the 𝑥-component of our first vector and the 𝑥-component of our second vector. And similarly, the 𝑦-component of our resultant vector is given by the sum of the 𝑦-component of the first vector and the 𝑦-component of the second vector.
Looking back to the question, we see that we’re asked to find the length of the red vector, which we’re told is the resultant of the blue and green vectors in the diagram. And if we look at our diagram, we see that this red vector does indeed start at the tail of the blue vector and extend to the tip of the green vector. In this question, the blue vector is entirely horizontal and the green vector is entirely vertical. This means that the angle between these two vectors is 90 degrees. This makes our job simpler. Since we know that our red resultant vector starts at the tail of the blue vector and extends to the tip of the green vector, and now we know that the angle between the blue and the green vectors is 90 degrees, then we know that our three vectors must 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 this right-angled triangle. If we label the lengths of the sides of this triangle as 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. If we take the square root of both sides of this equation, then we can rewrite this as 𝑐, the length of the hypotenuse, equals the square root of 𝑎 squared plus 𝑏 squared.
Now, in our case, 𝑎 and 𝑏 are the lengths of the blue vector and the green vector, respectively, and 𝑐 is the length of the red vector, the resultant vector that we are trying to find. So what this equation is saying is that in order to find the length of this resultant, we need to know the lengths 𝑎 and 𝑏 of the blue vector and the green vector. Luckily for us, we have a scale on our diagram. And since the blue vector and the green vector both point along the lines in this diagram, that makes it easy to read off their lengths. We know that each square is one centimeter in length. This means that if we count how many squares each of these vectors extends, this number is equal to the length of that vector in centimeters.
For our green vector, this is really straightforward because we’ve got the ruler positioned in the direction that the green vector is pointing. If we look at the tail of this green vector and we trace across to our ruler, we see that it starts at a height of zero centimeters. And if we look at the tip of this vector, we see that hits at a height of 10 centimeters. So we can say that 𝑏 is equal to 10 centimeters.
If we now look at our blue vector, we see that we don’t have a ruler positioned in the diagram to just read off its length. So we do actually need to count the squares. If we do this, we find that our blue vector is equal to one, two, three, four, five, six, seven, eight, nine, 10 squares in length. And since we know that one square is equal to one centimeter in length, we can say that the length of our blue vector 𝑎 is equal to 10 centimeters.
So now we can take these values of 𝑎 and 𝑏 and substitute them in to our equation for 𝑐. Since 𝑎 and 𝑏 each equal 10 centimeters, then we have that 𝑐 is given by the square root of 10 centimeters squared plus 10 centimeters squared. Now, we need to take a little care with the units when doing this calculation, because if we calculate the square of 10 centimeters, we get 100 centimeters squared. If we add together 100 centimeters squared and 100 centimeters squared, we get a result of 200 centimeters squared.
The last step to calculate the value of 𝑐, which gives us the length of the resultant vector, is to evaluate this square root. If we take the square root of a quantity measured in centimeters squared, then we get a quantity 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. So this quantity here, 𝑐, that we have calculated is the length of the resultant vector that we were asked to find. But if we look back at the question, we see that we’re asked to give our answer to the nearest centimeter. So our result rounds to 14 centimeters.
And so, finally, we get our answer to the question that, to the nearest centimeter, the length, the resultant vector in the diagram, is 14 centimeters.