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Magazine Chapter 9: Problem 20
Find the magnitude oi each of the following vectors. a. \([1, i, i]\) b. \([1+i, 1-i, 1+i]\) c. \([1+i, 2+i, 3+i]\) d. \([i, 1+i, 1-i, i]\) e. \([1+i, 1-i, i, 1-i]\)
Short Answer
Expert verified
a. \(\sqrt{3}\), b. \(\sqrt{6}\), c. \(\sqrt{17}\), d. \(\sqrt{6}\), e. 3.
Step by step solution
01
Understand the Magnitude Formula for Complex Vectors
The magnitude of a complex vector \( \mathbf{v} = [v_1, v_2, \ldots, v_n] \) is given by \( ||\mathbf{v}|| = \sqrt{|v_1|^2 + |v_2|^2 + \cdots + |v_n|^2} \), where \( |v_i| \) is the modulus of the complex number \( v_i \). The modulus of a complex number \( a + bi \) is \( |a + bi| = \sqrt{a^2 + b^2} \).
02
Calculate Magnitude for Vector a
Vector \([1, i, i]\). Calculate each component's modulus: - \(|1| = \sqrt{1^2 + 0^2} = 1\) - \(|i| = \sqrt{0^2 + 1^2} = 1\)- \(|i| = \sqrt{0^2 + 1^2} = 1\)Use the magnitude formula: \(||[1, i, i]|| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}\).
03
Calculate Magnitude for Vector b
Vector \([1+i, 1-i, 1+i]\). Calculate each component's modulus: - \(|1+i| = \sqrt{1^2 + 1^2} = \sqrt{2}\) - \(|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}\)- \(|1+i| = \sqrt{1^2 + 1^2} = \sqrt{2}\)Use the magnitude formula: \(||[1+i, 1-i, 1+i]|| = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{6}\).
04
Calculate Magnitude for Vector c
Vector \([1+i, 2+i, 3+i]\). Calculate each component's modulus: - \(|1+i| = \sqrt{1^2 + 1^2} = \sqrt{2}\) - \(|2+i| = \sqrt{2^2 + 1^2} = \sqrt{5}\)- \(|3+i| = \sqrt{3^2 + 1^2} = \sqrt{10}\)Use the magnitude formula: \(||[1+i, 2+i, 3+i]|| = \sqrt{2 + 5 + 10} = \sqrt{17}\).
05
Calculate Magnitude for Vector d
Vector \([i, 1+i, 1-i, i]\). Calculate each component's modulus: - \(|i| = \sqrt{0^2 + 1^2} = 1\) - \(|1+i| = \sqrt{1^2 + 1^2} = \sqrt{2}\)- \(|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}\)- \(|i| = \sqrt{0^2 + 1^2} = 1\) Use the magnitude formula: \(||[i, 1+i, 1-i, i]|| = \sqrt{1^2 + (\sqrt{2})^2 + (\sqrt{2})^2 + 1^2} = \sqrt{6}\).
06
Calculate Magnitude for Vector e
Vector \([1+i, 1-i, i, 1-i]\). Calculate each component's modulus: - \(|1+i| = \sqrt{1^2 + 1^2} = \sqrt{2}\) - \(|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}\)- \(|i| = \sqrt{0^2 + 1^2} = 1\)- \(|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}\)Use the magnitude formula: \(||[1+i, 1-i, i, 1-i]|| = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2 + 1^2 + (\sqrt{2})^2} = \sqrt{9} = 3\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Magnitude Formula
In the world of vectors, the magnitude is a measure of how "long" the vector is. For complex vectors, we use a special formula to find this length. Consider a complex vector \( \mathbf{v} = [v_1, v_2, \ldots, v_n] \). The magnitude of this vector is calculated using:
- \( ||\mathbf{v}|| = \sqrt{|v_1|^2 + |v_2|^2 + \cdots + |v_n|^2} \)
Modulus of Complex Numbers
The modulus of a complex number measures its size or distance from the origin in the complex plane. Represent any complex number as \( a + bi \). To find its modulus:
- Use the formula \( |a + bi| = \sqrt{a^2 + b^2} \).
Vector Magnitude Calculation
Let's see how to calculate the magnitude for vectors with complex components using the steps provided in the textbook solutions. Consider the vector \([1+i, 2+i, 3+i]\):
- First, find the modulus of each component:
- \(|1+i| = \sqrt{1^2 + 1^2} = \sqrt{2}\)
- \(|2+i| = \sqrt{2^2 + 1^2} = \sqrt{5}\)
- \(|3+i| = \sqrt{3^2 + 1^2} = \sqrt{10}\)
- \( ||[1+i, 2+i, 3+i]|| = \sqrt{2 + 5 + 10} = \sqrt{17} \)
Complex Number Properties
Complex numbers have fascinating properties that impact how we work with vectors. A few important aspects to remember include:
- Complex Conjugate: For any complex number \( a + bi \), its conjugate is \( a - bi \). This is often used in calculations to rationalize denominators.
- Addition and Subtraction: Simply add or subtract the real parts and the imaginary parts separately. For example, \((1 + i) + (2 - i) = 3 + 0i = 3\).
- Modulus Consistency: The modulus of a conjugate is the same as the original complex number, i.e., \(|a + bi| = |a - bi|\).
