Calculating the Angle Between Two Vectors Given Their Resultant and Magnitudes

Understanding the relationship between the resultant of two vectors, their individual magnitudes, and the angle between them is a fundamental concept in vector mathematics. This article explores how to find the angle between two vectors when their resultant and the relationship between their magnitudes are known. We will delve into the mathematical steps required to solve this problem and understand the underlying principles.

Introduction to Vectors and Resultants

A vector is a mathematical object that has both magnitude and direction. The resultant of two vectors is the vector sum of these two vectors. This sum can be represented graphically by drawing the vectors head to tail and then drawing a line from the tail of the first vector to the head of the second vector. The length and direction of this line represent the resultant vector.

Problem Constraints and Steps

The problem at hand specifies the following conditions:

The resultant of two vectors is 3. One vector is double the magnitude of the other vector.

Let’s denote the smaller vector by A with magnitude k and the larger vector by B with magnitude 2k. We need to find the angle θ between these two vectors.

Solution Using the Law of Cosines

The Law of Cosines is a powerful tool for solving problems involving the sides and angles of a triangle. In this case, we can apply it to the triangle formed by the two vectors and their resultant. The cosine rule states:

c2 a2 b2 - 2abcos(θ)

Here, c is the magnitude of the resultant, and a and b are the magnitudes of the vectors. In our problem, we have:

c 3 a k b 2k

Substituting these values into the law of cosines, we get:

32 k2 (2k)2 - 2k(2k)cos(θ)

This simplifies to:

9 k2 4k2 - 4k2cos(θ)

9 5k2 - 4k2cos(θ)

Solving for cos(θ) gives:

cos(θ) (5k2 - 9) / 4k2

The angle θ can be found by taking the arccosine (inverse cosine) of the above expression:

θ arccos[(5k2 - 9) / 4k2]

The Supplement of the Angle

The angle we have calculated, θ, is the angle opposite the resultant. However, we are usually interested in the angle between the two vectors, which is the supplement of θ. Therefore:

Angle between vectors 180° - θ

Verification and Simplification

For the equation 5k2 - 9 4k2 to hold true, we solve for k2:

5k2 - 9 4k2

5k2 - 4k2 9

k2 9

k 3

Substituting k 3 into the angle formula, we get:

θ arccos[(5(3)2 - 9) / 4(3)2]

θ arccos[(45 - 9) / 36]

θ arccos[36 / 36]

θ arccos(1)

Since arccos(1) 0° or 0 radians, the angle between the two vectors is 180° - 0° 180°.

Conclusion

In summary, the angle between two vectors when the resultant is 3 and one vector is double the other can be calculated using the Law of Cosines. This problem involves understanding vector magnitude, resultant vectors, and the application of trigonometric principles. The solution clearly demonstrates the importance of precise mathematical manipulation and understanding the geometric interpretation of vectors.

Further Reading and Exploration

For a deeper dive into vector mathematics, explore topics such as vector addition, scalar multiplication, and vector projection. Understanding these concepts will enhance your ability to solve more complex problems involving vectors.

Keywords: vector resultant, angle between vectors, vector magnitude