Understanding the Angle Between Vectors When One is Twice Another

The relationship between vectors, particularly when one vector is twice another, has long puzzled mathematicians and physicists. When the resultant vector of two such vectors is larger than both, determining the angle between them can be quite challenging. This article will explore this concept, providing a comprehensive breakdown of the problem and offering solutions for different scenarios.

Introduction to Vectors and Vector Addition

Vectors are mathematical entities that possess both magnitude and direction. The addition of vectors is essential in many scientific and engineering applications. The resultant vector is the result of adding two or more vectors together. In this scenario, we will discuss the situation where one vector is twice as long as another, and their resultant is larger than both.

Case Studies and Calculations

Scenario 1: When the Resultant Vector is Equal to the Larger Vector

In this case, if vector A is twice the vector B, and the resultant vector A B is equal to the larger vector, we can visualize a triangle with sides as A, B, and the resultant vector. The formula to find the angle between A and B can be derived from the law of cosines. This law states:

c2 a2 b2 - 2ab cos(y)

Where c is the resultant vector, a and b are the vectors, and y is the angle between them.

Given that A is twice B, we can represent the relationship as follows:

a 2b

According to the given information, the resultant vector (c) is equal to the larger vector, which can be represented as:

c 2b

Therefore, substituting the values in the law of cosines:

(2b)2 (2b)2 b2 - 2(2b)(b) cos(y)

4b2 4b2 b2 - 4b2 cos(y)

b2 4b2 cos(y)

cos(y) 1/4

y arccos(1/4)

Here, y is the angle between vector A and B.

Scenario 2: When the Resultant Vector is Larger Than Both Vectors

If the resultant vector is larger than both vectors, the vectors must be pointing in the same direction more than usual. To solve this, we use the formula derived from the parallelogram law of vector addition:

(A B)2 A2 B2 2AB cos(y)

Given that A is twice B, we can represent the relationship as:

(2B B)2 (2B)2 B2 2(2B)(B) cos(y)

9B2 4B2 B2 4B2 cos(y)

9B2 - 5B2 4B2 cos(y)

4B2 4B2 cos(y)

cos(y) -1/4

y arccos(-1/4)

In this case, the angle y is greater than arccos(1/4).

Key Points and Clarifications

1. Identical Vectors: A common misconception is that identical vectors are always equal. This is true if they are also identical in direction and magnitude. In vector spaces, vectors can be identical but point in different directions or have different magnitudes. The norm of a vector (its length) plays a crucial role in determining angles.

2. Vector Space and Norms: A vector space can be equipped with a norm (length) and an inner product (dot product). The angle between two vectors is only defined in the context of an inner product space. If the norms of the vectors add up to the norm of the resultant vector, it suggests that the vectors are aligned in the same direction.

Conclusion

Understanding the angle between vectors when one is twice another involves a detailed examination of the vectors' magnitudes and directions. By utilizing the law of cosines and the parallelogram law of vector addition, we can derive the relationship between the vectors and their resultant. The angle between the vectors depends on their relative directions and magnitudes, and further clarification is necessary when dealing with identical, unit, or parallel vectors.

Keywords

vector addition, resultant vector, angle between vectors, vector space