Understanding Vector Magnitude and Direction: A Comprehensive Guide

When working with vectors, it is essential to understand how to determine their magnitude and direction. This article will walk you through a scenario involving a vector where the horizontal component is double the vertical component. We will explore how to calculate the magnitude and direction of such a vector.

Scenario Overview

Let's denote the vertical component of the vector as V and the horizontal component as H. According to the problem, the horizontal component is double the vertical component, which can be expressed as:

H 2V

Magnitude of the Vector

The magnitude M of the vector can be calculated using the Pythagorean theorem:

M sqrt{H^2 V^2}

Substituting H with 2V:

M sqrt{(2V)^2 V^2} sqrt{4V^2 V^2} sqrt{5V^2} Vsqrt{5}

Direction of the Vector

The direction angle theta of the vector with respect to the horizontal axis can be found using the tangent function:

tan theta fraction{V}{H}

Substituting H with 2V:

tan theta fraction{V}{2V} fraction{1}{2}

To find theta, take the arctangent:

theta tan^{-1}left(fraction{1}{2}right)

This gives us:

theta 26.57^{circ}

Thus, the vector has a magnitude of Vsqrt{5} and a direction of approximately 26.57^{circ} from the horizontal axis.

Additional Scenario

Another scenario involves doubling both the horizontal and vertical components of a vector. In this case, the magnitude of the resultant vector is also doubled, but its direction remains unchanged. This is comparable to an enlarged photograph, where both the length and height can double, but the overall appearance remains the same.

Calculation Verification

If we call the vertical component magnitude Y, then the horizontal component can be written in terms of Y based on the given information. Now we have the two leg lengths in terms of Y. This means we can write the tangent of the angle of the resultant formed by adding the two components. Using the definition of tangent, we can find the direction angle. To find the magnitude of the resultant, we use the Pythagorean theorem. Our answer will be in terms of Y since we don't know that value.

Conclusion

In summary, a vector with a horizontal component double its vertical component has a magnitude of Vsqrt{5} and a direction of approximately 26.57^{circ} from the horizontal axis. Doubling both components of a vector doubles its magnitude but leaves its direction unchanged. Understanding these principles is crucial for various applications in mathematics, physics, and engineering. Whether it's working with enlargements or analyzing complex vector scenarios, the concepts discussed here provide a solid foundation.