Vector Representation

Definition

Vector representation refers to the mathematical way of expressing a quantity or entity that has both magnitude and direction. It is a fundamental concept in various fields, including physics, engineering, and computer science, and is particularly relevant in the context of complex numbers and their polar form.

5 Must Know Facts For Your Next Test

1. In the context of complex numbers, the vector representation of a complex number $z = a + bi$ is given by the ordered pair $(a, b)$, where $a$ is the real part and $b$ is the imaginary part.
2. The magnitude (or modulus) of a complex number $z = a + bi$ is given by the formula $|z| = \\sqrt{a^2 + b^2}$, which represents the length of the vector.
3. The argument (or angle) of a complex number $z = a + bi$ is given by the formula $\theta = \tan^{-1}\left(\frac{b}{a}\right)$, which represents the angle of the vector with respect to the positive real axis.
4. The polar form of a complex number $z = a + bi$ is given by $z = |z|\left(\cos\theta + i\sin\theta\right)$, where $|z|$ is the magnitude and $\theta$ is the argument of the complex number.
5. Vector representation of complex numbers is useful for performing operations such as addition, subtraction, multiplication, and division, as well as for visualizing the behavior of complex numbers on the Argand diagram.

Review Questions

• Explain how the vector representation of a complex number is related to its polar form.
  • The vector representation of a complex number $z = a + bi$ is given by the ordered pair $(a, b)$, which corresponds to the coordinates of the vector on the Argand diagram. The polar form of the same complex number is given by $z = |z|\left(\cos\theta + i\sin\theta\right)$, where $|z|$ is the magnitude (or modulus) of the vector, and $\theta$ is the argument (or angle) of the vector with respect to the positive real axis. The vector representation and the polar form are closely related, as the magnitude and argument of the complex number can be directly derived from the vector representation using the formulas $|z| = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1}\left(\frac{b}{a}\right)$.
• Describe how the vector representation of complex numbers can be used to perform operations such as addition and multiplication.
  • The vector representation of complex numbers is particularly useful for performing operations such as addition and multiplication. To add two complex numbers $z_1 = a_1 + b_1i$ and $z_2 = a_2 + b_2i$, represented as vectors $(a_1, b_1)$ and $(a_2, b_2)$, respectively, we can simply add the corresponding components: $(a_1 + a_2, b_1 + b_2)$. This corresponds to the geometric interpretation of adding the vectors tip-to-tail. To multiply two complex numbers $z_1 = a_1 + b_1i$ and $z_2 = a_2 + b_2i$, represented as vectors $(a_1, b_1)$ and $(a_2, b_2)$, we can use the formula $z_1 \times z_2 = (a_1a_2 - b_1b_2) + (a_1b_2 + a_2b_1)i$, which corresponds to the geometric interpretation of multiplying the magnitudes and adding the angles of the vectors.
• Analyze the relationship between the vector representation of complex numbers and their graphical representation on the Argand diagram.
  • The vector representation of a complex number $z = a + bi$ is closely tied to its graphical representation on the Argand diagram. The vector $(a, b)$ corresponds to the point on the complex plane where the real part $a$ is plotted on the horizontal axis, and the imaginary part $b$ is plotted on the vertical axis. The length of the vector, given by $|z| = \sqrt{a^2 + b^2}$, represents the magnitude or modulus of the complex number. The angle of the vector with respect to the positive real axis, given by $\theta = \tan^{-1}\left(\frac{b}{a}\right)$, represents the argument or angle of the complex number. This connection between the vector representation and the Argand diagram visualization allows for a deeper understanding of the properties and operations of complex numbers, as well as their geometric interpretation.

Related terms

Complex Number: A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, satisfying the equation $i^2 = -1$.

Polar Coordinates: Polar coordinates are a two-dimensional coordinate system that uses the distance from a reference point (the magnitude or modulus) and the angle from a reference direction (the argument or angle) to locate a point on a plane.

Argand Diagram: An Argand diagram is a graphical representation of the complex plane, where the real part of a complex number is plotted on the horizontal axis and the imaginary part is plotted on the vertical axis.