A person rode a bicycle seven root two kilometers east and then he rode for another
21 kilometers 45 degrees south of east. Determine the magnitude and direction of the displacement, rounding the direction to
the nearest minute.
The diagram shows the journey of the person from point 𝐴 to point 𝐶. He firstly rode due east seven root two kilometers. He then rode 21 kilometers 45 degrees south of east. The magnitude of the displacement is the distance 𝑥 from 𝐴 to 𝐶 and the direction
of the displacement is the angle 𝜃.
As angles on a straight line add up to 180 degrees, we know that the angle 𝐴𝐵𝐶 is
equal to 135 degrees. We can now use the cosine rule — 𝐴 squared equals 𝐵 squared plus 𝐶 squared minus
two 𝐵𝐶 cos 𝐴 — to calculate the magnitude of the displacement.
Substituting in our values from the diagram gives us 𝑥 squared is equal to 21
squared plus seven root two squared minus two multiplied by 21 multiplied by seven
root two multiplied by cos 135. This gives us a value of 𝑥 squared equal to 833. Square rooting both sides of this equation gives us 𝑥 is equal to seven root 17. Therefore, the magnitude of the displacement is seven root 17 kilometers.
In order to work out the direction of the displacement — the angle 𝜃 in this case —
we will use the sine rule. sin 𝐴 divided by 𝐴 is equal to sin 𝐵 divided by
𝐵. Substituting in the values from the diagram gives us sin 𝜃 divided by 21 is equal to
sine of 135 divided by seven root 17. We can multiply both sides of this equation by 21 so that sin 𝜃 is equal to sine of
135 divided by seven root 17 multiplied by 21. This gives us a value of 0.5145.
In order to calculate 𝜃, we do inverse sine or sine to the minus one of this
answer. Therefore, our angle 𝜃 is 30.964 degrees. As we want our answer to the nearest minute, we need to turn 0.964 into minutes. We do this by multiplying by 60 as there are 60 minutes in one degree. 0.964 multiplied by 60 is equal to 57.84. Therefore, our angle to the nearest minute is 30 degrees and 58 minutes.
This means that the direction of the displacement to the nearest minute is 30 degrees
and 58 minutes south of east.