Question

An aeroplane flies 150 miles on a bearing of \(105^{\circ}\) and then 107 miles on a bearing of \(217^{\circ}\). Find the bearing that the aeroplane must take to fly directly back to the starting position.

Short answer

Answer: To find the bearing angle, follow these steps: 1. Understand the problem. 2. Represent the information as vectors. 3. Calculate the vectors. 4. Find the required vector. 5. Calculate the coordinates of the vector. 6. Find the bearing angle using the arctangent formula and the coordinates of the required vector.

Step-by-step solution

Understanding the Problem

We are given that the aeroplane flies 150 miles on a bearing of \(105^{\circ}\) and then 107 miles on a bearing of \(217^{\circ}\). We need to find the bearing that the aeroplane must take to fly directly back to the starting position.

Representing the Information as Vectors

We can represent the initial 150 miles on a bearing of \(105^{\circ}\) as a vector in the plane as follows. Let A be the starting point, \(B_1\) be the point after flying 150 miles towards \(105^\circ\) and \(B_2\) be the final point after flying 107 miles towards \(217^\circ\). We need to find the bearing when flying back from the point \(B_2\) to A.

Calculating the Vectors

The vector from A to \(B_1\) can be represented as: \(\vec{AB_1} = 150\begin{bmatrix}\cos(105^\circ)\\\sin(105^\circ)\end{bmatrix}\). The vector from \(B_1\) to \(B_2\) can be represented as: \(\vec{B_1B_2} = 107\begin{bmatrix}\cos(217^\circ)\\\sin(217^\circ)\end{bmatrix}\). Adding the two vectors, we get the resultant vector from A to \(B_2\): \(\vec{AB_2} = \vec{AB_1} + \vec{B_1B_2}\).

Finding the Required Vector

To fly back directly to the starting position from \(B_2\), the required vector will be the negative of the resultant vector from A to \(B_2\): \(\vec{B_2A} = -\vec{AB_2}\).

Calculating the Coordinates of the Vector

Now calculate the coordinates of the vector \(\vec{AB_2}\) using the previously calculated vectors as: \(\vec{AB_2} = 150\begin{bmatrix}\cos(105^\circ)\\\sin(105^\circ)\end{bmatrix} + 107\begin{bmatrix}\cos(217^\circ)\\\sin(217^\circ)\end{bmatrix}\). Compute the result to obtain the vector \(\vec{AB_2}\).

Finding the Bearing Angle

Finally, we need to find the angle that represents the required bearing. To do this, use the arctangent to find the angle between the horizontal axis and the vector \(\vec{B_2A}\): \(\theta = \arctan{\frac{y}{x}}\), where \(x\) and \(y\) are the coordinates of the vector \(\vec{B_2A}\). Let's not forget to add 360 degrees or subtract 360 degrees if necessary to make the anglepositive and to stay between 0 and 360 degrees, i.e., in the first or fourth quadrant. Now we have the angle \(\theta\), which represents the bearing that the aeroplane must take to fly directly back to the starting position.

Key concepts

Bearing Calculation

Bearings are crucial in navigation as they determine the direction of travel. A bearing is typically measured in degrees, starting from the north direction, moving clockwise. For instance, a bearing of \(0^{\circ}\) would point directly north, while \(90^{\circ}\) would indicate east. In the context of the exercise, the plane's initial path is drawn on a bearing of \(105^{\circ}\). This direction can be used to visualize a path slightly east of southeast. When calculating bearings, drawing a diagram helps to conceptualize the angles and directions involved. The bearing difference between two points indicates how much one needs to turn clockwise from one direction to reach another. It's essential to ensure that all bearings remain within the \(0^{\circ}\) to \(360^{\circ}\) range. Any angle adjustments needed, such as adding or subtracting \(360^{\circ}\), keep the angle valid within this circular range.

Vector Addition

Vector addition is a fundamental concept in physics and engineering that helps to find a resultant direction or force. Each segment of the plane’s journey can be converted into vectors. For instance, the path from point A to \(B_1\) is represented as \(\vec{AB_1}\). In this exercise, vector addition involves adding the vector \(\vec{AB_1}\) and \(\vec{B_1B_2}\), each representing a leg of the flight. This is achieved computationally by adding the respective components of the vectors:

• Calculate the x and y components separately.
• Use trigonometric functions like cosine for the x component and sine for the y component.

The resultant vector \(\vec{AB_2}\) gives the comprehensive path the plane has traveled. Reversing this gives \(\vec{B_2A}\)—the vector representing the direct path back to the start.

Arctangent Function

The arctangent function, often denoted as \( \arctan \), is used to find an angle with a given tangent value. It’s particularly useful in navigation for calculating bearing or direction based on the components of a vector.When we have a vector in components with coordinates \((x, y)\), the bearing (or direction) back from \(B_2\) to the starting point is calculated using:\[ \theta = \arctan\left(\frac{y}{x}\right) \]This function returns an angle whose tangent is the quotient \(\frac{y}{x}\).The computed angle \(\theta\) usually lies between \(-90^{\circ}\) and \(90^{\circ}\), but when considering directions, adjustments may be needed:

• Ensure \(\theta\) fits into a \(0^{\circ}\) to \(360^{\circ}\) range by adding \(360^{\circ}\) if negative.
• Account for the quadrant in which the vector coordinates reside, as this influences the final bearing direction.

These manipulations ensure accurate bearing calculations, allowing navigators to interpret directions relative to the north reference.