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.