a plane flies due east for 500 km. and then turns on a heading of 120 degrees for 150 km. What are its distance and bearing from its starting point?
Add the two vectors. I assume that "heading" is measured clockwise from north.
They should have made that clear.
To find the distance and bearing from the starting point, we can use vector addition and trigonometry.
First, let's draw a diagram to visualize the plane's movements. Start by drawing a straight line to represent the initial eastward flight of 500 km. Then, draw a line at a 120-degree angle from the end of the first line to represent the second leg of the flight, which is 150 km long.
Now, we need to break down the second leg into its eastward and northward components. To do this, we can use trigonometry. Since the angle between the eastward direction and the second leg is 120 degrees, the eastward component can be found by calculating the cosine of 120 degrees and multiplying it by the length of the second leg (150 km). Similarly, the northward component can be found by calculating the sine of 120 degrees and multiplying it by the length of the second leg (150 km).
Eastward component = cos(120°) * 150 km
Northward component = sin(120°) * 150 km
Now, let's calculate these components.
Eastward component = (cos(120°)) * 150 km
= (-0.5) * 150 km
= -75 km (negative because it's to the west)
Northward component = (sin(120°)) * 150 km
= (0.866) * 150 km
= 129.9 km
To find the bearing angle, we can use the inverse tangent (arctan) function, which gives us the angle between the east direction and the resultant vector (total movement). The bearing angle will be measured counterclockwise from the east direction.
Bearing angle = arctan(Northward component / Eastward component)
Bearing angle = arctan(129.9 km / -75 km)
= arctan(-1.732)
Since the tangent function is negative in the second quadrant, we need to add 180 degrees to the calculated bearing angle.
Bearing angle = arctan(-1.732) + 180 degrees
= -60 degrees + 180 degrees
= 120 degrees
So, the distance from the starting point is the magnitude of the resultant vector, which can be found using the Pythagorean theorem.
Distance = sqrt((500 km)^2 + (Eastward component)^2 + (Northward component)^2)
= sqrt((500 km)^2 + (-75 km)^2 + (129.9 km)^2)
Calculating this value will give us the distance from the starting point.