- Convert the compass bearing to a standard Cartesian angle using \(\theta = 90^\circ - \text{bearing}\), resulting in \(240^\circ\).
- Use the magnitude and the Cartesian angle to define the vector: \(\mathbf{r} = \langle 350 \cos 240^\circ, 350 \sin 240^\circ \rangle\).
- Calculate the numerical components: \(x = -175\) and \(y \approx -303.11\).
- The final component form is \(\mathbf{r} = \boxed{\langle -175, -303.11 \rangle}\).
Explanation
- Analysis of the Jet's Motion and Coordinate Systems
We are given that a jet is flying at a constant speed of \(350\) miles/hour. The direction is given as a compass bearing of \(210^\circ\), where \(90^\circ\) represents North. To find the component form of the velocity vector \(\mathbf{r} = \langle x, y \rangle\), we need to determine its magnitude and its angle \(\theta\) in the standard Cartesian coordinate system, where \(0^\circ\) represents East and angles increase counter-clockwise. The relationship between the vector components and these parameters is given by \(\mathbf{r} = \langle |\mathbf{r}| \cos \theta, |\mathbf{r}| \sin \theta \rangle\).
- Determining the Cartesian Angle and Vector Components
In the standard Cartesian system, the compass bearing \(\alpha\) (measured clockwise from North) is related to the Cartesian angle \(\theta\) (measured counter-clockwise from East) by the formula \(\theta = 90^\circ - \alpha\). Given \(\alpha = 210^\circ\), we have \(\theta = 90^\circ - 210^\circ = -120^\circ\). To express this as a positive angle within the interval \([0^\circ, 360^\circ)\), we add \(360^\circ\): \(\theta = -120^\circ + 360^\circ = 240^\circ\). Thus, the vector representing the jet's motion is \(\mathbf{r} = \langle 350 \cos 240^\circ, 350 \sin 240^\circ \rangle\).
- Calculation and Matching Values
We calculate the components of the vector:
- The x-component is \(x = 350 \cos 240^\circ = 350 \times (-0.5) = -175\).
- The y-component is \(y = 350 \sin 240^\circ \approx 350 \times (-0.866025) \approx -303.11\).
Comparing these results with the provided values, we identify that the jet's motion can be represented as \(\mathbf{r} = \langle 350 \cos 240^\circ, 350 \sin 240^\circ \rangle = \langle -175, -303.11 \rangle\).
Examples
This type of vector calculation is essential in aviation for flight planning and air traffic control. By representing the velocity as a vector, pilots and systems can account for wind speed and direction by adding wind vectors to the jet's velocity vector to determine the resulting ground speed and track. Understanding these concepts helps ensure that planes arrive safely at their destinations by maintaining the correct flight path relative to the ground.