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Magazine Chapter 4: Problem 233
Prove that if the sum of three unit vectors is equal to \(\overrightarrow{0}\), then the angle between each pair of these vectors is equal to \(120^{\circ}\).
Short Answer
Expert verified
The angle between each pair of the vectors is \(120^{\circ}\).
Step by step solution
01
Define the unit vectors
Let the three unit vectors be \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) such that \( \mathbf{a} + \mathbf{b} + \mathbf{c} = \overrightarrow{0} \). This implies that \( \mathbf{c} = - (\mathbf{a} + \mathbf{b}) \).
02
Use the property of dot product
Given that they are unit vectors, we know \( \mathbf{a} \cdot \mathbf{a} = \mathbf{b} \cdot \mathbf{b} = \mathbf{c} \cdot \mathbf{c} = 1 \). Also, using their sum \( \mathbf{a} + \mathbf{b} + \mathbf{c} = \overrightarrow{0} \), we deduce \( \mathbf{c} = - (\mathbf{a} + \mathbf{b}) \).
03
Expand dot products using vector equality
Evaluate \( (\mathbf{a} + \mathbf{b} + \mathbf{c}) \cdot (\mathbf{a} + \mathbf{b} + \mathbf{c}) = 0 \), which simplifies to \( \mathbf{a} \cdot \mathbf{a} + \mathbf{b} \cdot \mathbf{b} + \mathbf{c} \cdot \mathbf{c} + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}) = 0 \). Since each vector's dot product with itself is 1, the equation becomes \( 3 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}) = 0 \).
04
Solve for the dot product
From the equation \( 3 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}) = 0 \), solve for the sum of dot products: \( \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a} = -\frac{3}{2} \).
05
Find the angle using dot product property
For any two vectors \( \mathbf{u} \) and \( \mathbf{v} \), the dot product \( \mathbf{u} \cdot \mathbf{v} = \| \mathbf{u} \| \| \mathbf{v} \| \cos \theta \). Since all vectors are unit vectors, \( \mathbf{a} \cdot \mathbf{b} = \cos \theta \), \( \mathbf{b} \cdot \mathbf{c} = \cos \theta \), and \( \mathbf{c} \cdot \mathbf{a} = \cos \theta \).
06
Equate each dot product to a common angle
Since each dot product equals to \( \cos \theta = -\frac{1}{2} \), it follows that \( \theta = 120^{\circ} \) as \( \cos 120^{\circ} = -\frac{1}{2} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Unit Vectors
In vector geometry, a unit vector is a vector that has a magnitude of 1. These vectors are often used to indicate direction, as their magnitude remains constant at one. They play a crucial role in vector operations because they simplify calculations:
- Unit vectors can be used to express any given vector, by scaling the unit vector by the vector's magnitude.
- A common notation for unit vectors in three-dimensional space includes \( \hat{i}, \hat{j}, \hat{k} \) which represent the axes of the coordinate system.
- If \( \mathbf{v} \) is a vector, then the unit vector \( \hat{v} \) in the direction of \( \mathbf{v} \) is given by \( \hat{v} = \frac{\mathbf{v}}{\|\mathbf{v}\|} \), where \( \|\mathbf{v}\| \) is the magnitude of vector \( \mathbf{v} \).
Dot Product
The dot product, also referred to as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number, which can be used to determine various properties of vectors relating to their relative orientation:
- The formula for the dot product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is \( \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta \), where \( \theta \) is the angle between the vectors.
- When vectors are in standard position (like unit vectors), it simplifies to \( \mathbf{u} \cdot \mathbf{v} = \cos \theta \) as their magnitudes are 1.
- The dot product is zero if the vectors are perpendicular (\( \theta = 90^{\circ} \)).
- If the dot product is positive, the angle between the vectors is less than \( 90^{\circ} \); if negative, the angle is more than \( 90^{\circ} \).
Angle Between Vectors
The angle between two vectors is crucial for understanding their relative orientation. Using the dot product is a fundamental way to calculate this angle, particularly when working with unit vectors:
- Given two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the angle \( \theta \) formed between them can be calculated using the dot product formula: \( \theta = \cos^{-1}\left(\frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|}\right) \).
- For unit vectors, this formula simplifies to \( \theta = \cos^{-1}(\mathbf{a} \cdot \mathbf{b}) \), as their magnitudes are 1.
- Understanding the angle between vectors is crucial in many areas, such as physics for understanding force directions and in engineering for determining stresses and strains.
