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Complex Numbers: Definition and Vector Form Video Lecture | Mathematics (Maths) for JEE Main & Advanced
FAQs on Complex Numbers: Definition and Vector Form Video Lecture - Mathematics (Maths) for JEE Main & Advanced
| 1. What are complex numbers and how are they defined? |  |
Ans. Complex numbers are numbers that can be expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, defined by the property \( i^2 = -1 \). The real part of the complex number is \( a \), and the imaginary part is \( b \).
| 2. How can we represent complex numbers in vector form? | |
Ans. Complex numbers can be represented in vector form as \( (a, b) \), where \( a \) is the real part and \( b \) is the imaginary part. This representation corresponds to a point or a vector in a two-dimensional Cartesian coordinate system, where the x-axis represents the real part and the y-axis represents the imaginary part.
| 3. What is the geometric interpretation of complex numbers? | |
Ans. The geometric interpretation of complex numbers involves plotting them on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. Each complex number corresponds to a point in this plane, and operations like addition and multiplication can be visualized as geometric transformations.
| 4. How do we add and subtract complex numbers using their vector form? | |
Ans. To add two complex numbers \( z_1 = a + bi \) and \( z_2 = c + di \) in vector form \( (a, b) \) and \( (c, d) \), respectively, we simply add their corresponding components: \( z_1 + z_2 = (a+c) + (b+d)i \). For subtraction, we subtract the components: \( z_1 - z_2 = (a-c) + (b-d)i \).
| 5. What are the modulus and argument of a complex number? | |
Ans. The modulus of a complex number \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \), which represents the distance from the origin to the point \( (a, b) \) in the complex plane. The argument, denoted as \( \arg(z) \), is the angle \( \theta \) formed with the positive x-axis, calculated using \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).
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