Problem 65 - Trigonometry

Question

Approximate the magnitude of each vector and the angle \(\theta, 0^{\circ} \leq \theta<360^{\circ}\), that the vector makes with the positive \(x\)-axis. Round your answers to the nearest tenth. $$\mathbf{W}=\sqrt{5} \mathbf{i}-2 \mathbf{j}$$

Step-by-Step Solution

Answer

Magnitude: 3, Angle: 318.2°

Step 1: Write the vector in component form

The vector \( \mathbf{W} \) is given as \( \mathbf{W} = \sqrt{5} \mathbf{i} - 2 \mathbf{j} \). This corresponds to the components \( W_x = \sqrt{5} \) and \( W_y = -2 \).

Step 2: Calculate the magnitude of the vector

The magnitude of the vector \( \mathbf{W} \) is calculated using the formula: \[ |\mathbf{W}| = \sqrt{W_x^2 + W_y^2} = \sqrt{(\sqrt{5})^2 + (-2)^2} = \sqrt{5 + 4} = \sqrt{9} = 3. \] Thus, the magnitude is \( 3 \).

Step 3: Calculate the angle with the positive x-axis

To find \( \theta \), use the formula \( \theta = \tan^{-1} \left( \frac{W_y}{W_x} \right) \). Substitute the values: \( \theta = \tan^{-1} \left( \frac{-2}{\sqrt{5}} \right) \). Calculate this using a calculator: \( \theta \approx \tan^{-1}(-0.8944) \approx -41.8^{\circ} \).

Step 4: Adjust the angle to fit the range \(0^{\circ} \leq \theta < 360^{\circ}\)

The angle \( -41.8^{\circ} \) is less than \( 0^{\circ} \) and needs to be adjusted by adding \( 360^{\circ} \). Therefore, \( \theta = 360^{\circ} - 41.8^{\circ} = 318.2^{\circ} \). This angle is now within the specified range.

Key Concepts

Understanding the Magnitude of a Vector

The magnitude of a vector provides a way to describe its size or length in space. For any vector with components, such as \[ \mathbf{W} = \sqrt{5} \mathbf{i} - 2 \mathbf{j}, \]we can find the magnitude using the formula: \[ |\mathbf{W}| = \sqrt{W_x^2 + W_y^2}, \]where \( W_x \) and \( W_y \) are the components along the x and y axes.

Considering the components:

• \( W_x = \sqrt{5} \)
• \( W_y = -2 \)

substituting into the formula gives: \[ \sqrt{(\sqrt{5})^2 + (-2)^2} = \sqrt{5 + 4} = \sqrt{9} = 3. \]This shows the vector \( \mathbf{W} \) has a magnitude of 3, which means it stretches 3 units in the plane. This calculation uses Pythagorean theorem principles, helping to show how vector length relates to its components.

Calculating the Angle a Vector Makes with the x-axis

The angle a vector makes with the positive x-axis can tell us about its direction in the coordinate plane. To find this angle \( \theta \), we need to understand how the vector points relative to the x-axis. This involves using the arctangent function, \( \tan^{-1} \), which can calculate angles based on given ratios.

The formula is: \[ \theta = \tan^{-1}\left(\frac{W_y}{W_x}\right). \]In our case:

• \( W_x = \sqrt{5} \)
• \( W_y = -2 \)

When plugged into the formula, it results in: \[ \theta = \tan^{-1}\left(\frac{-2}{\sqrt{5}}\right) = \tan^{-1}(-0.8944). \]Using a calculator, we find \( \theta \approx -41.8^\circ \). Since the angle needs to be positive, add 360 degrees, leading to \( \theta = 318.2^\circ \). This corrected angle indicates a rotation counterclockwise from the x-axis to the vector's direction.

Trigonometric Calculations for Vectors

Trigonometry plays a vital role in vector analysis, especially when determining angles or directional properties. Trigonometric functions, such as sine, cosine, and tangent, allow us to translate vector components into meaningful angles.

Here's how they come into play:

• **Tangent Function**: \( \frac{\text{opposite}}{\text{adjacent}} \), reflecting the ratio needed to find \( \theta \).
• **Arctangent Function**: \( \tan^{-1} \) is used to find an angle from a tangent ratio.

Using these functions:- To find \( \theta \), we apply \( \tan^{-1} \). For vectors, \( \tan^{-1}\left(\frac{W_y}{W_x}\right) \) offers the direction clue.

Properly adjusting angles is crucial when the vector's components change direction (like negative \( y \)-values). Adjustments by adding or subtracting full rotations ensure the angle remains within a standard 0 to 360-degree range. Understanding these trigonometric details assists in comprehensively analyzing vectors and their orientations.