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, in this question, we’re given a scale diagram that has three vectors in it. We’re told that the red vector is the resultant of the blue and green vectors. We’re also told that the sides of the squares in this diagram are one centimeter long, 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’s 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. Then drawing two vectors tip to tail means drawing the second vector with its tail at the tip of the first vector like this.
Then we can get the sum of these two vectors, which is their resultant vector, by drawing an arrow that starts at the tail of the first vector and extends to the tip of the second vector. So in this example, that blue arrow that we’ve just added to the diagram is our resultant vector. Now that we have seen what is meant by a resultant vector, let’s look back at the question. We are told that the red vector is the resultant of the blue and the green vectors. And if we look at the diagram, we see that the blue vector and the green vector are drawn tip to tail; that is, the green vector is drawn starting with its tail at the tip of the blue vector.
Looking now at the red vector, we see that it’s drawn with its tail at the tail of the first vector, the blue vector, and with its tip at the tip of the second vector, the green vector. This means that the red vector is indeed the resultant of the blue vector and the green vector. In this question, the blue vector is entirely horizontal and the green vector is entirely vertical. This means that we know that the angle between these two vectors is equal to 90 degrees. In other words, we can say that three vectors in the diagram form a right-angled triangle in which the red resultant vector is the hypotenuse of the triangle. Since the question is asking us to find the length of the resultant vector, this means that we need to find the length of this hypotenuse of the triangle.
Since we have a right-angle triangle, let’s recall Pythagoras’s theorem. If we label the lengths of the sides of our 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. To answer this question, we’re trying to find the length of the hypotenuse or the value of 𝑐. So let’s take the square root of both sides of this equation in order to make 𝑐 the subject. Taking the square root, we have that 𝑐 is equal to the square root of 𝑎 squared plus 𝑏 squared. This means that in order to find the value of 𝑐, 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. The question tells us that the sides of the squares of this grid are one centimeter long. 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 we have a square grid, then if each square is one centimeter in length in the vertical direction, then each square must also be one centimeter in length in the horizontal direction. So one square’s worth of distance in either the horizontal direction or the vertical direction corresponds to a one centimeter.
This means that in order to find the values of 𝑎 and 𝑏, all we need to do is start at the tail of each vector and count the number of squares until we reach the tip of this vector. That number of squares then gives the length of the vector measured in centimeters. Let’s begin with the blue vector. We start at the tail of this vector and we 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 equal to eight. So we can say that the length of the blue vector and the value of 𝑎 is equal to eight centimeters.
Now, looking at the green vector, we’ll start at the tail of that vector, which is at the tip of the blue vector, and count the number of squares until we reach the tip. In this case, we find that that number of squares is 10. And so we can say that the length of the green vector and the value of 𝑏 is equal to 10 centimeters. Now that we have values for both 𝑎 and 𝑏, we can substitute them into this equation for 𝑐. Substituting in that 𝑎 equals eight centimeters and 𝑏 equals 10 centimeters, we get that 𝑐 is equal to the square root of eight centimeters squared plus 10 centimeters squared.
At this point, we need to take some care with our units. When we take the square of a quantity with units of centimeters, we’re going to get a quantity with units of centimeters squared. In this case, taking the square of eight centimeters gives us 64 centimeters squared. And taking the square of 10 centimeters gives us 100 centimeters squared. Adding 64 centimeters squared to 100 centimeters squared, we get 164 centimeters squared. Then all that’s left to do to get our value of 𝑐 is to evaluate the square root.
Now, if we take the square root of a quantity with units of centimeters squared, we’ll get a result with units of centimeters. Now, of course, this makes sense because 𝑐 should have units of distance. After all, it is a length. It’s the length of the hypotenuse of this triangle or, equivalently, the length of this red vector in our diagram. If we take the square root of 164, we get a result of 12.806 and so on with further decimal places. And this result here for 𝑐 is the length of the resultant vector that the question was asking us to find. But if we look back at the question, we see that it asked us to give our answer to the nearest centimeter. So our value of 12.806 rounds up to give 13. And we have that 𝑐 is equal to 13 centimeters.
And so our answer to the question is that the length of the resultant vector, measured to the nearest centimeter, is equal to 13 centimeters.
