# Question Video: Finding the Magnitude of the Displacement between Two Points Physics • 9th Grade

Point 𝐴 is located 8 m horizontally from the base of the wall of a house, and point 𝐵 is located 6 m vertically above the base of the wall, as shown in the diagram. What is the magnitude of the displacement from point 𝐴 to point 𝐵?

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### Video Transcript

Point 𝐴 is located eight meters horizontally from the base of the wall of a house, and point 𝐵 is located six meters vertically above the base of the wall, as shown in the diagram. What is the magnitude of the displacement from point 𝐴 to point 𝐵?

Okay, so we can see that in this question, we’ve got a diagram of a house. And there are two points marked in this diagram, which are labeled 𝐴 and 𝐵. We are being asked to find the magnitude of the displacement from this point here, that’s point 𝐴, to point 𝐵, which is this point here. The displacement from point 𝐴 to point 𝐵 is represented by an arrow that starts at point 𝐴 and extends up to point 𝐵. So that’s an arrow with its tail at point 𝐴 and its tip at point 𝐵.

We can recall that displacement is a vector quantity, which means that it has both a magnitude and a direction. The direction of the displacement from point 𝐴 to point 𝐵 is the direction that the pink arrow is pointing in. The magnitude of this displacement is the length of the arrow, and it’s this magnitude that the question asks us to find. In order to do this, we can notice that the displacement vector that we’ve drawn from point 𝐴 to point 𝐵 forms the hypotenuse of a right-angled triangle. The other two sides of the triangle have lengths of eight meters and six meters.

Since the right-angled corner is at the base of the wall and we are told that point 𝐴 is a horizontal distance of eight meters from this base and that point 𝐵 is a vertical distance of six meters above it, to work out the magnitude of the displacement or the length of this hypotenuse, we can make use of Pythagoras’s theorem. This theorem tells us that if we have a right-angled triangle like this that has a hypotenuse of length 𝑐 and other sides of lengths 𝑎 and 𝑏 that 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared. Or, in words, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If we take the square root of both sides of this equation, we have that 𝑐, the length of the hypotenuse, is equal to the square root of 𝑎 squared plus 𝑏 squared.

By comparing this triangle that we’ve identified in our diagram to this general right-angled triangle here, we can see that the side marked as lowercase 𝑎 has a length of eight meters and that the side marked as lowercase 𝑏 has a length of six meters. So we’ve got our values for the quantities 𝑎 and 𝑏 in this equation. And if we substitute those values in, we can use the equation to calculate the value of 𝑐. And this value is the length of the hypotenuse of this triangle, which is equal to the magnitude of the displacement from point 𝐴 to point B.

Substituting these values in, we find that 𝑐 is equal to the square root of eight meters squared plus six meters squared. The square of eight meters plus the square of six meters works out as 100 meters squared. Finally, evaluating the square root gives a result of 10 meters. And so we have found that the magnitude of the displacement from point 𝐴 to point 𝐵 is 10 meters.