Determine the Angle Between Unit Vectors for a Given Condition

When dealing with vector mathematics, it is often necessary to determine the angle between unit vectors under specific conditions. This is particularly useful in physics and engineering applications where vector properties play a crucial role. In this article, we delve into the process of determining the angle between vectors vector a and vector b when 3a - b is a unit vector. Let's explore the solution step by step, adhering to the guidelines for effective SEO and SEOer best practices.

Introduction to Unit Vectors and Vector Operations

Unit vectors are vectors that have a magnitude of 1. They are often used to define directions in vector spaces. The properties of unit vectors make them particularly useful in simplifying complex vector problems. In vector mathematics, one of the operations commonly performed is the dot product, which is useful for finding the angle between two vectors. The dot product of two vectors vector a and vector b can be expressed as:

[ vector a cdot vector b |vector a||vector b|costheta ]

Where |vector a| and |vector b| are the magnitudes of the vectors, and theta is the angle between them.

The Problem: Given Vectors and Conditions

The problem at hand is to find the angle theta between unit vectors vector a and vector b such that 3a - b is a unit vector. Let's first express this condition mathematically:

[ ||3a - b|| 1 ]

To proceed, we will square both sides of the equation:

[ (3a - b) cdot (3a - b) 1^2 ]

Expanding the left-hand side using the distributive property of the dot product:

[ 9(a cdot a) - 6(a cdot b) (b cdot b) 1 ]

Since both vector a and vector b are unit vectors, (a cdot a) 1 and (b cdot b) 1. Substituting these values in, we get:

[ 9 - 6(a cdot b) 1 1 ]

Combining constants on the left-hand side:

[ 10 - 6(a cdot b) 1 ]

Isolating the dot product term:

[ 6(a cdot b) 9 ]

Dividing both sides by 6:

[ a cdot b frac{3}{2} ]

Given that vector a and vector b are unit vectors, the dot product can be expressed in terms of the cosine of the angle between them:

[ a cdot b costheta ]

Thus:

[ costheta frac{3}{2} ]

At this point, we need to check whether this value is possible for the cosine of an angle. The cosine of an angle can only range from -1 to 1. Since 3/2 is greater than 1, it is not possible for the cosine of an angle to be greater than 1. Therefore, we conclude that there is no solution to the given problem under the given conditions.

Alternative Scenarios and Analysis

Let's also consider other possible scenarios to explore further:

1. **If 3a - b is a unit vector, then 3a - b 1.

[ (3a - b) cdot (3a - b) 1 ]

Expanding the left-hand side:

[ 9(a cdot a) - 6(a cdot b) (b cdot b) 1 ]

Since a and b are unit vectors, (a cdot a) 1 and (b cdot b) 1. Substituting these values in:

[ 9 - 6(a cdot b) 1 1 ]

Combining constants:

[ 10 - 6(a cdot b) 1 ]

Isolating the dot product term:

[ 6(a cdot b) 9 ]

Dividing by 6:

[ a cdot b frac{3}{2} ]

Again, this value is not possible for the cosine of an angle, confirming our previous conclusion.

2. **The smallest possible value of 3a - b occurs when a b and 3a - b 3a - a 2a.

If a is a unit vector, 2a is not a unit vector since its magnitude is 2. Therefore, 3a - b cannot be a unit vector in this scenario.

3. **The smallest length is when 3a is in the same direction as b and that length is 2.

If 3a is in the same direction as b, then their difference will be 2, which is not a unit vector.

Therefore, the angle between the unit vectors vector a and vector b that satisfies the given condition is not possible based on the properties of unit vectors and vector operations.