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Magazine Chapter 12: Problem 24
Diagonal of parallelogram Show that the indicated diagonal of the parallelogram determined by vectors \(u\) and \(\mathbf{v}\) bisects the angle between \(\mathbf{u}\) and \(\mathbf{v}\) if \(|\mathbf{u}|=|\mathbf{v}|\).
Short Answer
Expert verified
The diagonal bisects the angle between \( \mathbf{u} \) and \( \mathbf{v} \) if \(|\mathbf{u}| = |\mathbf{v}|\).
Step by step solution
01
Understand the Geometric Set-Up
We are given a parallelogram with vectors \( \mathbf{u} \) and \( \mathbf{v} \) as adjacent sides. The diagonals of this parallelogram are created by the vectors \( \mathbf{u} + \mathbf{v} \) and \( \mathbf{u} - \mathbf{v} \). Given that \(|\mathbf{u}| = |\mathbf{v}|\), we need to show that the diagonal \( \mathbf{u} + \mathbf{v} \) bisects the angle between \( \mathbf{u} \) and \( \mathbf{v} \).
02
Vector Addition and Definition of Bisector
The diagonal \( \mathbf{u} + \mathbf{v} \) is a vector formed by adding \( \mathbf{u} \) and \( \mathbf{v} \). A bisector of an angle formed between two vectors is a vector which forms equal angles with each of the two original vectors. Therefore, we need to show that \( \mathbf{u} + \mathbf{v} \) makes equal angles with both \( \mathbf{u} \) and \( \mathbf{v} \).
03
Use of Dot Product
The dot product formula \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \) (where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \)) helps to find angles. We calculate \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{u}\) and \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{v}\).
04
Calculate Dot Products
Compute \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{u} = \mathbf{u} \cdot \mathbf{u} + \mathbf{v} \cdot \mathbf{u} = |mathbf{u}|^2 + \mathbf{v} \cdot \mathbf{u}\). Similarly, \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{v} = |mathbf{v}|^2 + \mathbf{u} \cdot \mathbf{v} \).
05
Compare the Dot Products
Since \(|\mathbf{u}| = |\mathbf{v}|\), \(|\mathbf{u}|^2 = |\mathbf{v}|^2\). Thus, \((\mathbf{u} + \mathbf{v}) \cdot \mathbf{u} = (\mathbf{u} + \mathbf{v}) \cdot \mathbf{v}\), which implies that the angle between \(\mathbf{u}\) and \(\mathbf{u} + \mathbf{v}\) is equal to the angle between \(\mathbf{v}\) and \(\mathbf{u} + \mathbf{v}\).
06
Conclusion
Since the diagonal \( \mathbf{u} + \mathbf{v} \) makes equal angles with both \( \mathbf{u} \) and \( \mathbf{v} \), it bisects the angle between them when \(|\mathbf{u}| = |\mathbf{v}|\). This satisfies the required condition, and the statement is proven.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Addition
Vector addition is a fundamental operation in vector spaces. It involves simply adding corresponding components of two or more vectors together. Imagine we have two vectors, \( \mathbf{u} \) and \( \mathbf{v} \). When we perform vector addition, we are essentially placing the tail of \( \mathbf{v} \) at the head of \( \mathbf{u} \) and drawing a new vector from the tail of \( \mathbf{u} \) to the head of \( \mathbf{v} \). This new vector, formed through addition, is represented as \( \mathbf{u} + \mathbf{v} \).
- Easy to visualize: Simply align the first vector and attach the second vector to the end of it.
- Coordinate-wise calculation: Add the components of the vectors separately, such as \( (x_1 + x_2, y_1 + y_2) \).
Dot Product
The dot product is a means of combining two vectors to get a scalar. It's calculated by multiplying corresponding components of vectors and then adding them up. For vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \), the dot product is given by:\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2.\]
- Measures angles: The dot product is used to find the cosine of the angle between vectors, using the formula \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \).
- Zero product: If the dot product is zero, the vectors are orthogonal (right-angled).
Angle Bisector
An angle bisector is a line or vector that splits an angle into two equal parts. In vector terms, a bisector between vectors means it makes equal angles with them. For the diagonal \( \mathbf{u} + \mathbf{v} \) in our case, ensuring that it bisects is about demonstrating that the angle between \( \mathbf{u} \) and \( \mathbf{u} + \mathbf{v} \) equals the angle between \( \mathbf{v} \) and \( \mathbf{u} + \mathbf{v} \).
- Equal contribution: It signifies that the diagonal is equally inclined towards both vectors \( \mathbf{u} \) and \( \mathbf{v} \).
- Symmetry: Balance between both vectors, which arises due to equal vector magnitudes.
Parallelogram Properties
Parallelograms are a special type of quadrilateral where opposite sides are parallel and equal in length. One of their distinctive features is the diagonals, which bisect each other. For a parallelogram formed by vectors \( \mathbf{u} \) and \( \mathbf{v} \), this means
- The two diagonals divide the parallelogram into two equal areas.
- This structure allows us to leverage vector addition and mid-point properties for geometric proofs.
