Understanding the Angle Between Unit Vectors: A Unit Vector Formation Condition

The concept of unit vectors and their manipulation through vector operations is a fundamental topic in advanced mathematics. One specific condition often explored in vector calculations is when a vector formed by combining two unit vectors is also a unit vector. This article delves into this specific scenario, specifically dealing with the equation √3a - b being a unit vector and the angle between the vectors mathbf{a} and mathbf{b}.

The Problem Statement

We are given two unit vectors, a and b, and we need to find their angle such that the vector √3a - b is also a unit vector. A unit vector has a magnitude of 1, and a vector is considered a unit vector if its magnitude is equivalent to 1. Therefore, the magnitude of the vector √3a - b must also be 1.

Mathematical Derivation

We start by setting up the condition for the magnitude of √3a - b to be 1:

[mathsqrt{3}mathbf{a} - mathbf{b} 1]

Calculating the magnitude:

[sqrt{(sqrt{3}mathbf{a} - mathbf{b})^2} 1]

Expanding this using the dot product:

[sqrt{sqrt{3}mathbf{a} - mathbf{b} cdot sqrt{3}mathbf{a} - mathbf{b}} 1]

Expanding further:

[sqrt{3mathbf{a}^2 - 2sqrt{3}mathbf{a} cdot mathbf{b} mathbf{b}^2} 1]

Simplifying using the fact that mathbf{a} and mathbf{b} are unit vectors (with magnitudes of 1):

[sqrt{3 - 2sqrt{3}mathbf{a} cdot mathbf{b} 1} 1]

This simplifies to:

[sqrt{4 - 2sqrt{3}mathbf{a} cdot mathbf{b}} 1]

Squaring both sides:

[4 - 2sqrt{3}mathbf{a} cdot mathbf{b} 1]

Rearranging to find mathbf{a} cdot mathbf{b}:

[2sqrt{3}mathbf{a} cdot mathbf{b} 3]

Dividing both sides by 2sqrt{3}:

[mathbf{a} cdot mathbf{b} frac{3}{2sqrt{3}} frac{sqrt{3}}{2}

Recall that the dot product mathbf{a} cdot mathbf{b} |mathbf{a}||mathbf{b}|cos{theta}. Since |mathbf{a}| |mathbf{b}| 1, this simplifies to:

[cos{theta} frac{sqrt{3}}{2}

Solving for the angle theta:

[theta 30^circ text{ or } 330^circ]

Conclusion

In conclusion, the angle between the unit vectors mathbf{a} and mathbf{b} such that √3a - b is a unit vector is 30 degrees or 330 degrees. This problem showcases the practical applications of vector operations and the conditions necessary for specific vector transformations.

Related Keywords

unit vectors dot product angle between vectors