Question

A horse trots 17 miles north and then turns and trots 24 miles at an angle of 40° west of north. Find the resultant displacement and direction mathematically. Hint: This is a non-perpendicular vector problem.

Answer 1

Resultant displacement = 38.58 miles at an angle of 23.57° north of west.

Step-by-step explanation:

Let north represent positive y direction and east represent positive x direction. Unit vector along x direction is represented by i and vector along unit vector along y direction is represented by j

Initial displacement = 17 miles north = 17 j miles

Final displacement = 24 miles at an angle of 40° west of north = 24 miles at an angle of 130° to positive x -axis

= (24 cos 130 i + 24 sin 130 j )

= (-15.43 i + 18.36 j ) miles

Total displacement = 17 j + (-15.43 i + 18.36 j) = -15.43 i + 35.36 j

Magnitude of displacement
`=√((-15.43)^2+35.36^2) =38.58 miles`

Angle to positive x -axis = tan⁻¹(35.36/-15.43) = 113.57°

Resultant displacement = 38.58 miles at an angle of 23.57° west of north.