Problem 63 - 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. $$\mathrm{U}=3 \mathrm{i}+5 \mathbf{j}$$

Step-by-Step Solution

Answer

The magnitude of \( \mathbf{U} \) is approximately 5.8, and the angle \( \theta \) is approximately \( 59.0^{\circ} \).

Step 1: Identify the components

Identify the components of vector \( \mathbf{U} = 3\mathbf{i} + 5\mathbf{j} \). The \( x \)-component is 3 and the \( y \)-component is 5.

Step 2: Calculate the magnitude

Use the formula for the magnitude of a vector \( \mathbf{U} = a\mathbf{i} + b\mathbf{j} \), given by \( \| \mathbf{U} \| = \sqrt{a^2 + b^2} \). Here, \( a = 3 \) and \( b = 5 \). Compute \( \| \mathbf{U} \| = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \).

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

Use the formula for the angle \( \theta \) given by \( \tan \theta = \frac{b}{a} = \frac{5}{3} \). Find \( \theta \) using \( \theta = \tan^{-1} \left( \frac{5}{3} \right) \). Calculate \( \theta \approx 59^{\circ} \).

Step 4: Round the results

Round the magnitude and the angle to one decimal place. The magnitude \( \sqrt{34} \approx 5.8 \) and the angle \( 59^{\circ} \) rounds to \( 59.0^{\circ} \).

Key Concepts

Vector Magnitude

Vectors are mathematical entities that have both magnitude and direction. The magnitude of a vector can be thought of as its length or size. Calculating the magnitude is a crucial step in understanding vectors, as it allows us to measure how much of something a vector represents regardless of direction.
To find the magnitude of a vector with components \( a \) and \( b \), such as \( \mathbf{U} = a\mathbf{i} + b\mathbf{j} \), you use the formula:

• \( \| \mathbf{U} \| = \sqrt{a^2 + b^2} \)

In our case:

• The \( x \)-component \( a = 3 \)
• The \( y \)-component \( b = 5 \)
• Putting it all together, \( \| \mathbf{U} \| = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \)

This calculation shows that the magnitude of vector \( \mathbf{U} \) is approximately \( 5.8 \) when rounded to one decimal place.

Angle with the X-Axis

Understanding how a vector is oriented in space involves calculating the angle it makes with the positive direction of the x-axis. This angle tells us the direction of the vector compared to a standard reference direction along the x-axis.
To find the angle \( \theta \), we use trigonometry, specifically the tangent function. The relation is:

• \( \tan \theta = \frac{b}{a} \)

For our vector \( \mathbf{U} = 3\mathbf{i} + 5\mathbf{j} \):

• \( \tan \theta = \frac{5}{3} \)

Using the inverse tangent function, also known as arctangent or \( \tan^{-1} \), we find:

• \( \theta = \tan^{-1} \left( \frac{5}{3} \right) \)

The calculated angle is approximately \( 59^{\circ} \). This means the vector \( \mathbf{U} \) is oriented \( 59^{\circ} \) above the positive x-axis.

Trigonometry Concepts

Trigonometry is essential for understanding vectors, especially when it comes to finding angles and distances. The primary trigonometric functions—sine, cosine, and tangent—relate the sides and angles of right triangles. In vector mathematics, these functions allow us to bridge the gaps between a vector's components and its overall direction and magnitude.

• Tangent: This function relates the opposite side to the adjacent side of a right triangle. In vectors, \( \tan \theta = \frac{ \text{opposite} }{ \text{adjacent} } = \frac{b}{a} \), helps us find the angle with the x-axis.
• Inverse Tangent: Also known as arctan or \( \tan^{-1} \), is used to retrieve the angle from the tangent value.
• These functions support converting between a vector's rectangular form, \( a\mathbf{i} + b\mathbf{j} \), and a more intuitive polar representation (magnitude \( |\mathbf{U}| \) and direction or angle \( \theta \)).

To sum it up, trigonometry provides the tools for resolving vectors into components and for converting these components back into meaningful physical directions, while maintaining a mathematically rigorous approach.