The summit of a mountain, 2270 above base camp, is measured on a map to be 4610 horizontally from the camp in a direction 30.9 west of north.What are the components of the displacement vector from camp to summit? Choose the axis east, axis north, and axis up.What is its magnitude?
x is east
y is north
z is up I assume
x = -4610 sin 30.9 East
y = +4610 cos 30.9 North
z = +2270
To find the components of the displacement vector from the camp to the summit, we need to break it down into its east, north, and up components.
First, let's consider the horizontal component, which is the displacement in the east direction. We can find it using trigonometry.
The given information states that the direction of the displacement vector is 30.9° west of north. However, to find the east component, we need the angle relative to the x-axis (east). So, we will subtract 30.9° from 90° to get the angle relative to the x-axis.
Angle relative to x-axis (east) = 90° - 30.9° = 59.1°
Now, we can use the given angle and the magnitude of the horizontal displacement to find the east component.
East component = Magnitude * cos(angle relative to x-axis)
East component = 4610 m * cos(59.1°)
Next, let's find the north component, which is the displacement in the north direction. We can also use trigonometry to calculate it.
North component = Magnitude * sin(angle relative to x-axis)
North component = 4610 m * sin(59.1°)
Lastly, we need to find the vertical component (up component). The given information states that the summit is 2270 m above the base camp. So, the up component is simply 2270 m.
Now, let's calculate the magnitude of the displacement vector using the components we found.
Magnitude = sqrt((East component)^2 + (North component)^2 + (Up component)^2)
Inserting the values we found above, we can calculate the magnitude of the displacement vector.