Question

What angle does \(\vec{A}=\left(A_{x}, A_{y}\right)=(30.0 \mathrm{~m},-50.0 \mathrm{~m})\) make with the positive \(x\) -axis? What angle does it make with the negative \(y\) -axis?

Short answer

Answer: The angle between vector A and the positive x-axis is \(301^{\circ}\), and the angle between vector A and the negative y-axis is \(149^{\circ}\).

Step-by-step solution

Find the angle between vector A and the positive x-axis

We have vector A with components \(A_x = 30.0 m\) and \(A_y = -50.0 m\). To find the angle between A and the positive x-axis, we use the arctangent function: \(\tan^{-1}\left(\frac{A_y}{A_x}\right)\) Calculate the arctangent of \(\frac{-50.0}{30.0}\), which can be done using a calculator: \(\theta_1 = \tan^{-1}\left(\frac{-50.0}{30.0}\right) = -59.0^{\circ}\)

Convert the angle to a positive value (if needed)

If the angle found in Step 1 is negative, add 360 degrees so that the angle represents a positive angle measured from the positive x-axis counterclockwise: \(\theta_1 = -59.0^{\circ} + 360^{\circ} = 301^{\circ}\) So, the angle between vector A and the positive x-axis is \(301^{\circ}\).

Find the angle between vector A and the negative y-axis

Now, we need to find the angle between vector A and the negative y-axis. To do this, simply subtract the angle found in Step 1 from 90 degrees: \(\theta_2 = 90^{\circ} - (-59.0^{\circ}) = 149^{\circ}\) The angle between vector A and the negative y-axis is \(149^{\circ}\).

Key concepts

Angle Calculation

Understanding how to calculate angles with vectors is essential in physics and engineering. When dealing with vectors, the angle can help determine direction and alignment with other components of the coordinate system. To calculate the angle between a vector and an axis, we typically rely on trigonometric functions, notably the arctangent function. This calculation involves the components of the vector, such as its position along the x and y axes, allowing us to determine the vector's orientation. When calculated correctly, this angle can also be adjusted to fit different quadrant systems of the standard plane, usually measured in degrees for easier interpretation.

Arctangent Function

The arctangent function, denoted as \tan^{-1}\, converts a ratio of two numbers into an angle. This function is instrumental in vector mathematics for finding the angle a vector makes with the axes. Specifically, it's used when you have the vector's components along the x and y axes.

• Formula: The angle \(\theta\) with the arctangent function is computed as \(\tan^{-1}\left(\frac{A_y}{A_x}\right)\).
• Sign: The angle often comes in a range of \(-90^{\circ}\) to \(90^{\circ}\). If negative, adjustments like adding \(360^{\circ}\) are made to convert it into a standard positive angle measure on a circle.

Understanding how to apply and adjust this result is crucial for accurate vector analysis. Using a calculator can simplify this process and help ensure the angle reflects the correct orientation.

Coordinate System

In vector mathematics, a coordinate system provides a context for analyzing directions and positions. It's essential when solving for angles, as it sets the foundation for how vector components are interpreted.

• 2D Planes: Most commonly, vectors are resolved within two-dimensional planes utilizing x and y coordinates, providing a frame of reference for direction.
• Quadrants: The plane is divided into four quadrants. These quadrants help determine the initial angle's sign and whether adjustments (like adding \(360^{\circ}\)) are needed.

Understanding the layout of your coordinate system can make a significant difference in interpreting and solving vector-based problems correctly.

Vector Components

Vectors are quantified by their components, which show a vector's impact along each axis of your coordinate system. Analysing these components is critical for tasks like angle calculation.

• X-component \(A_x\): The portion of the vector that runs parallel to the x-axis.
• Y-component \(A_y\): The segment of the vector paralleling the y-axis. Here, it was \(-50.0 m\), indicating a direction opposite the positive y.

These components (\(A_x = 30.0 m\) and \(A_y = -50.0 m\) in the example) offer a breakdown of the vector's overall direction and magnitude. By analyzing these values, you gain insight into how to adjust and compute angles and better understand the vector's role in resolving real-world physics problems.