Finding the Angle of a Vector in the XY Plane

In the field of vector analysis and trigonometry, vectors are fundamental concepts used to describe physical quantities with both magnitude and direction. A vector in the xy plane with a given magnitude and x-component presents an opportunity to explore trigonometric relationships. The following steps detail how to find the angle that a vector makes with the positive x-axis.

The Context

Consider a vector (mathbf{r}) in the xy plane with a magnitude of (29.0) and an x-component of (14.3). We aim to determine the angle (theta) that this vector makes with the positive x-axis.

Solution

The components of the vector are related to its magnitude through trigonometric functions. The cosine of the angle (theta) can be expressed as:

[cos theta frac{x}{r}]

Given that (r 29.0) and (x 14.3), we substitute these values into the formula:

[cos theta frac{14.3}{29.0}]

To solve for (theta), we take the inverse cosine (arccos or (cos^{-1})) of the fraction:

[theta cos^{-1}left(frac{14.3}{29.0}right)]

Using a calculator, we find:

[theta approx 60.455^circ]

Alternative Solution

Alternatively, the angle can be found in the fourth quadrant using:

[theta -cos^{-1}left(frac{14.3}{29.0}right)]

Calculating the Y-Component

First, let's find the y-component using the Pythagorean theorem:

[A_y sqrt{A^2 - A_x^2}]

Substitute (A 29.0) and (A_x 14.3):

[A_y sqrt{29^2 - 14.3^2}]

Note that:

[29^2 841, quad 14.3^2 204.49]

To simplify, perform the subtraction:

[29^2 - 14.3^2 841 - 204.49 636.51]

And take the square root:

[A_y sqrt{636.51} approx 25.23]

Evaluating the angle using the arctan function:

[theta tan^{-1}left(frac{A_y}{A_x}right)]

Substitute the values:

[theta tan^{-1}left(frac{25.23}{14.3}right)]

This yields:

[theta approx 60.455^circ]

Understanding Trigonometric Relations

For a right triangle, the relation between the adjacent side (x), the hypotenuse (r), and the angle (theta) is:

[cos theta frac{x}{r}]

This is a trigonometric function that describes the ratio of the adjacent side to the hypotenuse in a right triangle.

Conclusion

By utilizing the Pythagorean theorem and trigonometric identities, we can accurately determine the angle that a vector makes with the positive x-axis. Understanding these concepts is crucial in many areas of physics, engineering, and mathematics.