Problem 19 - Essential Calculus Early Transcendentals

Question

What is the angle between the given vector and the positive direction of the \(x\) -axis? What is the angle between the given vector and the positive direction of the \(x\) -axis? $$\mathbf{i}+\sqrt{3} \mathbf{j}$$

Step-by-Step Solution

Answer

The angle is \( 60^\circ \).

Step 1: Understanding the vector components

The given vector is \( \mathbf{v} = \mathbf{i} + \sqrt{3} \mathbf{j} \), which in component form is \( \mathbf{v} = (1, \sqrt{3}) \). This means the vector has an \( x \)-component of 1 and a \( y \)-component of \( \sqrt{3} \).

Step 2: Formula for the angle

The angle \( \theta \) between a vector \( \mathbf{v} = (a, b) \) and the positive \( x \)-axis is given by the formula: \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).

Step 3: Calculating the tangent value

For our vector \( (1, \sqrt{3}) \), substitute \( a = 1 \) and \( b = \sqrt{3} \) into the formula to calculate the tangent: \( \frac{b}{a} = \frac{\sqrt{3}}{1} = \sqrt{3} \).

Step 4: Finding the inverse tangent

Now, find \( \theta = \tan^{-1}(\sqrt{3}) \). The angle for which the tangent is \( \sqrt{3} \) is \( 60^\circ \) (or \( \frac{\pi}{3} \) radians).

Step 5: Result verification

Verify that the calculated angle belongs to the correct interval \([0, 180^\circ]\) by considering the vector's direction and quadrant. Since both components of the vector are positive, it lies in the first quadrant, confirming \( 60^\circ \) is correct.

Key Concepts

Vector Components

When understanding vectors, it's important to know how they express movement or direction in space. A vector is represented by its components along the x and y axes. In simpler terms, you can think of these components as the vector's shadows on the respective axes.
For the vector given in the problem, \(\mathbf{v} = \mathbf{i} + \sqrt{3} \mathbf{j}\), it has an x-component of 1 and a y-component of \(\sqrt{3}\).
This means:

• The vector moves 1 unit in the direction of the x-axis.
• It also moves \(\sqrt{3}\) units in the direction of the y-axis.

This information is crucial because these components help determine the angle the vector makes with the x-axis. Mastering vector components allow you to understand vector direction and magnitude much more intuitively.

Tangent Inverse

The tangent inverse, often denoted as \(\tan^{-1}\) or \(\arctan\), is a mathematical function used to find angles when you know the ratio of the opposite side to the adjacent side in a right triangle. This concept is especially helpful in vector calculations.
To find the angle between a vector and the x-axis, the formula \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\) is used. Here, \(a\) is the x-component and \(b\) is the y-component of the vector.
In our vector with components (1, \(\sqrt{3}\)):

• \(b/a = \sqrt{3}/1 = \sqrt{3}\)

Therefore, you calculate the angle \(\theta\) for which \(\tan(\theta) = \sqrt{3}\). This allows you to find that \(\theta\) roughly equals 60 degrees. Understanding how to use the tangent inverse is a key skill when working with right triangles and vectors.

Angle Measurement

Angle measurement is crucial when describing the position or rotation of vectors. Angles can be measured in both degrees and radians, where 360 degrees equals \(2\pi\) radians. For vector problems, you often work within the 0 to 180-degree range for angles relative to the positive x-axis.
In this example, we determined the angle using the tangent inverse, resulting in \(60^{\circ}\). This is the angle between our vector and the positive x-axis.
Always remember:

• 0 degrees points directly along the positive x-axis.
• 90 degrees points straight upwards.
• 180 degrees points directly opposite the positive x-axis.

Being familiar with the unit circle and common angles like 30, 45, 60, and 90 degrees can also help you quickly recognize results without needing a calculator.

First Quadrant

The coordinate plane is divided into four quadrants, and understanding them is essential in vector analysis. Vectors' positions regarding these quadrants give insight into the angle measurements and the expected results.
If both components of a vector are positive, as with our vector (1, \(\sqrt{3}\)), the vector is located in the first quadrant.
The first quadrant is identified by:

• The x-coordinate and y-coordinate are both positive.
• Angles measured from the positive x-axis to a vector in this quadrant are always between 0 and 90 degrees.

This coincides with our calculated \(60^{\circ}\) angle, confirming it's correct. Understanding which quadrant your vector resides in helps verify the calculated angles and avoid errors due to arithmetical issues.