Video Transcript
A body moved 28 meters due east and then 14 meters due north. Determine the body’s displacement, stating its direction to the nearest minute.
Let’s begin by considering the four main compass points as shown. We will let one unit in the easterly direction be equal to the unit vector 𝐢 and one unit in the northerly direction be the unit vector 𝐣. We are told that the body initially moves 28 meters due east. This is equal to the vector 28𝐢. The body then moves 14 meters due north, which is equal to 14𝐣. The displacement vector of the body is therefore equal to 28𝐢 plus 14𝐣.
We can calculate the magnitude of any vector by finding the sum of the squares of each of its components and then square rooting the answer. In this case, we have the square root of 28 squared plus 14 squared. This is equal to the square root of 980, which in turn simplifies to 14 root five. The body’s displacement is 14 root five meters.
We are also asked to work out the direction, so we need to calculate the angle 𝜃. We have a right triangle with opposite side equal to 14 and adjacent equal to 28. The tangent ratio states that tan 𝜃 is equal to the opposite over the adjacent. Substituting in our values, we have tan 𝜃 is equal to 14 over 28. The fraction simplifies to give us one-half. We can then take the inverse tangent of both sides of this equation such that 𝜃 is equal to the inverse tan of one-half.
Typing this into the calculator gives us 26.5650 and so on degrees. We are asked to give our answer to the nearest minute. So we need to multiply the decimal part of this answer by 60. This gives us an answer of 26 degrees and 34 minutes to the nearest minute. As this is the angle above the horizontal, we can say that the direction of the body is 26 degrees and 34 minutes north of east. If a body moves 28 meters due east and then 14 minutes due north, its displacement is 14 root five meters, 26 degrees and 34 minutes north of east.
