Question

An air traffic controller observes two airplanes approaching the airport. The displacement from the control tower to plane 1 is given by the vector a > , which has a magnitude of 220 km and points in a direction 32° north of west. The displacement from the control tower to plane 2 is given by the vector b > , which has a magnitude of 140 km and points 65° east of north. (a) Sketch the vectors a > , -b > , and d > = a > - b > . Notice that d > is the displacement from plane 2 to plane 1. (b) Use components to find the magnitude and the direction of the vector d > .

Answer 1

The vector d = a - b is obtained by subtracting the components of vector b from a, and then the magnitude and direction of d are found using the Pythagorean theorem and the arctangent function respectively.

Step-by-step explanation:

To find the magnitude and direction of the vector d = a - b, we need to break down vectors a and b into their components along the north-south and east-west axes. Since vector points 32° north of west, its western component is aw = 220 × cos(32°) and its northern component is an = 220 × sin(32°). Vector b, pointing 65° east of north, has a northern component of bn = 140 × cos(65°) and an eastern component of be = 140 × sin(65°).

To find the components of d, we subtract the components of b from those of a: dw = aw + (-be) and dn = an - bn. The magnitude of d is then found using the Pythagorean theorem, and the direction is found using the arctangent of the ratio between the north and west components.