Simple Nature - Light and Matter

one point and ends at another. Adding two ∆r vectors is interpretedas a trip with two legs: by computing the ∆r vector going from pointA to point B plus the vector from B to C, we find the vector thatwould have taken us directly from A to C.Calculations with magnitude and directionIf you ask someone where Las Vegas is compared to Los Angeles,she is unlikely to say that the ∆x is 290 km and the ∆y is 230 km,in a coordinate system where the positive x axis is east and the yaxis points north. She will probably say instead that it’s 370 kmto the northeast. If she was being precise, she might specify thedirection as 38 ◦ counterclockwise from east. In two dimensions, wecan always specify a vector’s direction like this, using a single angle.A magnitude plus an angle suffice to specify everything about thevector. The following two examples show how we use trigonometryand the Pythagorean theorem to go back and forth between the x-yand magnitude-angle descriptions of vectors.Finding magnitude and angle from components example 59⊲ Given that the ∆r vector from LA to Las Vegas has ∆x=290 kmand ∆y=230 km, how would we find the magnitude and directionof ∆r?⊲ We find the magnitude of ∆r from the Pythagorean theorem:|∆r| =√∆x 2 + ∆y 2= 370 kmWe know all three sides of the triangle, so the angle θ can befound using any of the inverse trig functions. For example, weknow the opposite and adjacent sides, sol / Example 59.−1 ∆yθ = tan∆x= 38 ◦ .Finding the components from the magnitude and angle example60⊲ Given that the straight-line distance from Los Angeles to LasVegas is 370 km, and that the angle θ in the figure is 38 ◦ , howcan the x and y components of the ∆r vector be found?⊲ The sine and cosine of θ relate the given information to theinformation we wish to find:cos θ = ∆x|∆r|sin θ = ∆y|∆r|Section 3.4 Motion In Three Dimensions 197