Finding the Angle Between Vectors AB and B-A Given a and b
The problem involves finding the angle between two vectors, AB and B-A, given the magnitudes of vectors A and B and the angle between them. This article will walk through the steps and provide a detailed explanation to solve such problems.
Introduction
When dealing with vectors, one of the essential skills is to calculate the angles between vectors, especially under the given conditions. The problem discussed in this article involves the vectors A and B, which make an angle of x π/6. With A √3 and B 1, we need to find the angle between AB and B-A.
Step-by-Step Solution
Solution with Specific Values
Given: A √3, B 1, and the angle between A and B is π/6. Let's use these values to find the angle between vectors AB and B-A.
1. Calculate vector AB:
AB (3/2 - √3/2, 10 - 1)
2. Calculate vector B-A:
B-A (-1/2 - √3/2, 1 - √3)
3. Use the dot product to find the angle between AB and B-A:
AB · B-A (3/2 - √3/2)(-1/2 - √3/2) (10 - 1)(1 - √3) -8
4. Calculate the magnitudes of AB and B-A:
|AB| √(3^2 10^2 - 2 * √3 * 10 * cos(π/6))
|B-A| √((-1/2 - √3/2)^2 (1 - √3)^2)
5. Use the cosine formula to find the angle between AB and B-A:
cos(θ) (AB · B-A) / (|AB| * |B-A|)
θ arctan(sqrt(3)/5) π/2 arctan(sqrt(3)/5) 2π/30.20943951 ≈ 2.3038346 rad
General Solution Using Trigonometry
For a general case where |A| a, |B| b, and the angle between A and B is x:
The angle between vectors A and B can be represented in a geometric setup with vectors A and B drawn from a common origin. The magnitude of the resultant vectors AB and B-A can be found using the properties of vector addition and subtraction.
Step 1: Sketch the Vectors
Starting at point O, vector A points to point A, vector AB points to P, and vector B-A points to Q.
PAQ is a straight line of length 2, OA is length 3, and angles OAP π - π/6, OAQ π/6.
Step 2: Use the Law of Sines
The angle POQ can be found using the sine rule, which states that the ratio of the length of a side to the sine of the opposite angle is the same for all three sides of a triangle.
sin(angle POA) / |AB| sin(angle POQ) / |B-A|
angle POA angle POQ angle OAP π
angle POA angle POQ (π - π/6) π
angle POA angle POQ π/6
Step 3: Use Dot Product to Find the Angle Between AB and B-A
The dot product of the vectors AB and B-A can be used to find the cosine of the angle between them:
AB · B-A b2 - a2 12 - 32 -8
Then, use the cosine formula to find the angle:
cos(θ) (AB · B-A) / (|AB| * |B-A|)
θ arctan(b-a * sin(x) / b * a * cos(x))
Conclusion
The angle between vectors AB and B-A, given the magnitudes of A and B and the angle between them, can be found using vector addition, subtraction, and trigonometric identities. This solution demonstrates the application of vector algebra and trigonometry to solve practical problems involving vector angles.