Problem 35 - Trigonometry

Question

Draw vectors representing the course of a ship that travels $$ 25 \text { miles on a course with heading } 225^{\circ} $$

Step-by-Step Solution

Answer

Draw a vector at 135° from the x-axis with magnitude 25.

Step 1: Understand the Problem

We need to draw a vector that represents the course of a ship traveling 25 miles at a heading of 225°. Vectors are graphical representations of magnitude and direction. The magnitude in this case is 25 miles, and the direction is given by the angle 225°.

Step 2: Identify the Coordinate System

In navigation, headings are typically measured clockwise from the north. However, vectors on a Cartesian plane are measured counterclockwise from the positive x-axis. Thus, we need to adjust the angle to fit the Cartesian coordinate plane.

Step 3: Convert the Heading to Cartesian Angle

Since the heading of 225° is measured clockwise from the north, we need to convert this to the typical angle measurement from the positive x-axis. The positive y-axis is at 90° relative to the x-axis, so heading 225° from the north translates to 225° - 90° = 135° counterclockwise from the positive x-axis.

Step 4: Determine the Components of the Vector

We can break the vector into its horizontal (x) and vertical (y) components using trigonometry. The x-component can be found using \(x = 25 \cos(135°)\) and the y-component using \(y = 25 \sin(135°)\).

Step 5: Calculate the Components

Calculate the x and y components: \(x = 25 \cos(135°) = -25 / \sqrt{2}\approx -17.68\), \(y = 25 \sin(135°) = 25 / \sqrt{2}\approx 17.68\).

Step 6: Draw the Vector

Draw the vector with an initial point at the origin (0,0) extending to the point (-17.68, 17.68), since the x-component is negative and the y-component is positive.

Key Concepts

Vector Components

When we talk about vector components, we refer to breaking down a vector into its fundamental horizontal and vertical parts. This is crucial when dealing with problems involving magnitude and direction. By understanding vector components, you can easily determine the vector's action in the horizontal and vertical directions separately. Each vector component is essentially a vector in its own right, lying along either the x-axis or the y-axis in a Cartesian coordinate system.

To find the components of a vector, we use simple trigonometry. Imagine you have a vector making an angle with the x-axis. You can determine its components by considering its magnitude. For the vector moving 25 miles at an angle of 135°, you would break it down as follows:

• The horizontal component (x) is calculated using: \( x = ext{magnitude} \times ext{cos(angle)} \). For our exercise: \( x = 25 \times ext{cos(135°)} \).
• The vertical component (y) uses the sine function: \( y = ext{magnitude} \times ext{sin(angle)} \). Here, \( y = 25 \times ext{sin(135°)} \).

By finding these components, you effectively simplify the process of analyzing and visualizing the vector's path on the plane.

Coordinate System Conversion

Coordinate system conversion is a vital skill when dealing with vectors, especially in navigation and trigonometry tasks. Different applications can use different reference points for vectors, so converting angles and magnitudes into a single system is important for consistency.

In the problem, we're asked to convert a navigation heading into an angle appropriate for a Cartesian plane. Navigation headings are measured clockwise from due north, which is unlike the counterclockwise measurement from the positive x-axis standard in Cartesian coordinates. To convert, think of the positive y-axis (which is north) as 90° away from the x-axis. Consequently, a 225° heading means:

• You're moving 225° clockwise from north.
• This translates to a typical geometry angle of 225° - 90° = 135° counterclockwise from the positive x-axis.

Converting this angle ensures you're properly orienting your vector on a Cartesian graph, making further calculations feasible and accurate.

Magnitude and Direction

The magnitude and direction are the fundamental characteristics of any vector, describing exactly how far and in which direction the vector acts. Vectors are represented graphically as arrows, where the arrow length indicates magnitude, and the arrow's direction shows the vector's orientation relative to a reference point.

Let's revisit our exercise: the ship's movement is a vector of 25 miles in length at a heading of 225°. Here, '25 miles' is the magnitude of the vector, indicating how far the ship travels. The angle of 225° gives us the direction.

Visualize the vector by drawing a line from an initial point (like the origin (0,0) on a graph) with a length proportional to the magnitude (25 miles in this case), oriented 135° counterclockwise from the positive x-axis due to our coordinate system conversion. Understanding and applying these principles of magnitude and direction is key in accurately depicting the vector's reality on a plot or map.