Understanding the Relationship Between Vectors and Their Resultant

When working with vectors, one of the fundamental concepts is understanding how different vectors interact with each other. Specifically, if the magnitude of one vector is twice that of another, how do we find the angle between these vectors and the angle formed by the resultant of these vectors? This article aims to clarify these concepts and provide a step-by-step guide on finding the required relationships.

The Basics of Vectors and Their Magnitudes

Let's consider two vectors, a and b. Suppose the magnitude of vector a is twice that of vector b, denoted as:

a 2b

This means that the magnitude of vector a is directly proportional to twice that of b. With this understanding, we can now explore the relationship between the included angle between these vectors and the angle formed by their resultant vector.

Calculating the Magnitude of the Resultant Vector

The magnitude of the resultant vector R can be derived using the law of cosines. If the angle between a and b is denoted as θ, then the magnitude of the resultant vector R is given by:

R √(a2 b2 2ab cos θ)

Given that a 2b, we can substitute this into the equation:

R √((2b)2 b2 2(2b)(b) cos θ)

Simplifying further:

R √(4b2 b2 4b2 cos θ)

R √(5b2 4b2 cos θ)

Thus, the magnitude of the resultant vector R is:

R b √(5 4 cos θ)

Deriving the Resultant Angle

To find the angle between the resultant vector R and one of the individual vectors, say a, we need to use vector projection and trigonometry. Let's denote the angle between the resultant vector R and a as φ. Then:

tan φ (Projection of b on R) / (Magnitude of a)

The projection of vector b on R can be found using the dot product and the cosine of the angle between R and b. The dot product of vectors R and b is given by:

R · b |R| |b| cos φ

Using the law of cosines again, we can express |R| in terms of b and θ:

R · b (b √(5 4 cos θ)) · b cos φ b2 √(5 4 cos θ) cos φ

The projection of b on R is given by:

Projection of b on R (b · R) / |R| (b2 √(5 4 cos θ) cos φ) / (b √(5 4 cos θ))

Therefore, the projection simplifies to:

Projection of b on R b cos φ

Thus, we have:

tan φ (b cos φ) / (2b)

tan φ (cos φ) / (2)

Finally, solving for φ:

tan φ (1/2)

φ tan^(-1)(1/2)

Conclusion

In summary, given a scenario where the magnitude of one vector is twice that of another, the relationship between the included angle between these vectors and the resultant angle of these two vectors can be derived using vector projection and trigonometry. The key insights include:

The magnitude of the resultant vector is given by R b √(5 4 cos θ). The angle between the resultant vector and one of the individual vectors, if the magnitude of one vector is twice that of the other, is given by φ tan^(-1)(1/2).

This analysis clarifies the relationship between the given vectors and their resultant vector, providing insights for further study in the field of vector algebra.