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Magazine Chapter 17: Problem 38
Prove analytically the triangle inequality for vectors \(|\mathbf{A}+\mathbf{B}| \leq|\mathbf{A}|+|\mathbf{B}|\).
Short Answer
Expert verified
The triangle inequality for vectors is proved as \(|\textbf{A} + \textbf{B}| \leq |\textbf{A}| + |\textbf{B}|\).
Step by step solution
01
- Express Magnitude of Vectors
For vectors \(\textbf{A}\) and \(\textbf{B}\), the magnitudes are given by \(|\textbf{A}| = \sqrt{\textbf{A} \cdot \bf{A}}\) and \(|\textbf{B}| = \sqrt{\textbf{B} \cdot \textbf{B}}\).
02
- Consider the Magnitude of the Sum of Vectors
The magnitude of the sum of two vectors is \(|\textbf{A} + \textbf{B}| = \sqrt{(\textbf{A} + \textbf{B}) \cdot (\textbf{A} + \textbf{B})}\).
03
- Expand the Dot Product
Expand the dot product inside the square root: \[(\textbf{A} + \textbf{B}) \cdot (\textbf{A} + \textbf{B}) = \textbf{A} \cdot \textbf{A} + 2(\textbf{A} \cdot \textbf{B}) + \textbf{B} \cdot \textbf{B}\]
04
- Use the Cauchy-Schwarz Inequality
By the Cauchy-Schwarz inequality, \(\textbf{A} \cdot \textbf{B} \leq |\textbf{A}| |\textbf{B}|\). Thus, \[(\textbf{A} + \textbf{B}) \cdot (\textbf{A} + \textbf{B}) \leq \textbf{A} \cdot \textbf{A} + 2|\textbf{A}| |\textbf{B}| + \textbf{B} \cdot \textbf{B}\]
05
- Sum of Magnitudes
Given that \(\textbf{A} \cdot \textbf{A} = |\textbf{A}|^2\) and \(\textbf{B} \cdot \textbf{B} = |\textbf{B}|^2\), we get \[(\textbf{A} + \textbf{B}) \cdot (\textbf{A} + \textbf{B}) \leq |\textbf{A}|^2 + 2|\textbf{A}| |\textbf{B}| + |\textbf{B}|^2 = (|\textbf{A}| + |\textbf{B}|)^2\]
06
- Apply Square Roots
Taking the square root on both sides yields \(|\textbf{A} + \textbf{B}| \leq |\textbf{A}| + |\textbf{B}|\), thus proving the triangle inequality for vectors.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Magnitudes
In vector algebra, the magnitude (or length) of a vector \(\textbf{A}\) is a measure of how long the vector is. To represent the magnitude, we use \(|\textbf{A}|\). The formula to calculate the magnitude of a vector is \(|\textbf{A}| = \sqrt{\textbf{A} \cdot \textbf{A}}\).
Here, the dot product \(\textbf{A} \cdot \textbf{A}\) means you multiply each component of \(\textbf{A}\) by itself and then sum them all up.
For example, if \(\textbf{A} = (a_1, a_2, ..., a_n)\), then \(\textbf{A} \cdot \textbf{A} = a_1^2 + a_2^2 + ... + a_n^2\).
Taking the square root of this sum gives you the vector's magnitude.
Here, the dot product \(\textbf{A} \cdot \textbf{A}\) means you multiply each component of \(\textbf{A}\) by itself and then sum them all up.
For example, if \(\textbf{A} = (a_1, a_2, ..., a_n)\), then \(\textbf{A} \cdot \textbf{A} = a_1^2 + a_2^2 + ... + a_n^2\).
Taking the square root of this sum gives you the vector's magnitude.
Dot Product
The dot product of two vectors is a way to multiply them together to get a scalar value (a single number). For vectors \(\textbf{A}\) and \(\textbf{B}\), the dot product is denoted by \(\textbf{A} \cdot \textbf{B}\).
The formula for the dot product of \(\textbf{A} = (a_1, a_2, ..., a_n) \) and \(\textbf{B} = (b_1, b_2, ..., b_n) \) is:
When the dot product is zero, the vectors are perpendicular (at a right angle to each other).
The formula for the dot product of \(\textbf{A} = (a_1, a_2, ..., a_n) \) and \(\textbf{B} = (b_1, b_2, ..., b_n) \) is:
- \(\textbf{A} \cdot \textbf{B} = a_1b_1 + a_2b_2 + ... + a_nb_n \)
When the dot product is zero, the vectors are perpendicular (at a right angle to each other).
Cauchy-Schwarz Inequality
The Cauchy-Schwarz inequality is a critical tool in vector algebra. It states that for any vectors \(\textbf{A}\) and \(\textbf{B}\), the absolute value of their dot product is less than or equal to the product of their magnitudes: \(|\textbf{A} \cdot \textbf{B}| \leq |\textbf{A}| |\textbf{B}|\).
This inequality is useful for proving many other important properties of vectors, like the triangle inequality.
In our context, we use it to show that \(\textbf{A} \cdot \textbf{B} \leq |\textbf{A}| |\textbf{B}|\), which is an intermediate step in the proof of the triangle inequality for vectors.
This inequality is useful for proving many other important properties of vectors, like the triangle inequality.
In our context, we use it to show that \(\textbf{A} \cdot \textbf{B} \leq |\textbf{A}| |\textbf{B}|\), which is an intermediate step in the proof of the triangle inequality for vectors.
Vector Algebra
Vector algebra is a branch of mathematics that deals with vectors and the various operations you can perform on them.
Understanding these basic operations is crucial for deeper concepts like the triangle inequality.
- Addition of vectors: Given two vectors \(\textbf{A}\) and \(\textbf{B}\), their sum is another vector \(\textbf{C}\) such that \(\textbf{C} = \textbf{A} + \textbf{B}\)
- Scalar multiplication: If \(\textbf{A}\) is a vector and \(k\) is a scalar, then \(\textbf{B} = k\textbf{A}\)
- Dot product and cross product: ways to multiply vectors to get scalar or another vector, respectively
Understanding these basic operations is crucial for deeper concepts like the triangle inequality.
