An air plane flies 20 km in a direction 60 degree west of north then 40 km straight east and then 10 km in a direction 30 degree north of east . How far and at what direction is the plane from the starting point?
Disp. = 20km[150o]+40km[0o]+10km[30o].
X = 20*Cos150+40+10*Cos30 = 31.3 km.
Y = 20*sin150+10*sin30 = 15 km.
Tan A = Y/X = 15/31.3 = 0.47923.
A = 25.6o = Direction.
14+25+30
To find the distance and direction of the airplane from the starting point, we can use vector addition.
First, we need to convert the given information into vectors. We will use a coordinate system where North is the positive Y-direction, East is the positive X-direction, and the origin is the starting point.
Let's break down the airplane's movement into three vectors:
1. The first leg of the journey: The airplane flies 20 km in a direction 60 degrees west of North. We can break this down using trigonometry. The vertical component would be 20 km times the sine of 60 degrees, and the horizontal component would be 20 km times the cosine of 60 degrees. This gives us a vector of (-20 * sin(60), 20 * cos(60)) km.
2. The second leg of the journey: The airplane flies 40 km straight East, which means its vector would be (40, 0) km.
3. The third leg of the journey: The airplane flies 10 km in a direction 30 degrees North of East. Again, we can use trigonometry to break this down. The vertical component would be 10 km times the sine of 30 degrees, and the horizontal component would be 10 km times the cosine of 30 degrees. This gives us a vector of (10 * cos(30), 10 * sin(30)) km.
To find the total vector, we add up the three vectors:
Total vector = (-20 * sin(60) + 10 * cos(30) + 40, 20 * cos(60) + 10 * sin(30))
Now, we can find the magnitude (distance) and direction (angle) of the total vector using trigonometry.
Magnitude:
The magnitude of the total vector can be found using the Pythagorean theorem:
Magnitude = sqrt((x-coordinate)^2 + (y-coordinate)^2)
Direction:
The direction of the total vector can be found using the arctan function. The angle (theta) can be calculated as:
theta = arctan(y-coordinate / x-coordinate)
Now that we have the magnitude and direction, we can find the distance and direction of the airplane from the starting point.