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An Argand diagram is a tool to visualize complex numbers geometrically. On this diagram, the horizontal axis represents the real part of the complex number and the vertical axis represents the imaginary part. Each complex number corresponds to a unique point on the plane, much like a coordinate pair in geometry.
For example, the complex number \(3 + \mathrm{j}2\) is represented by the point (3, 2) on the Argand diagram, where 3 is the real component and 2 is the imaginary component. Similarly, \(-1 + \mathrm{j}4\) is the point (-1, 4).
- Real part: moves along the x-axis (horizontal line)- Imaginary part: moves along the y-axis (vertical line)
Using these points, geometric shapes like lines, triangles, and squares can be represented and manipulated. In our example, we can see the square ABCD with vertices at these points, visualizing vector movement and complex arithmetic.

Vectors in the context of complex numbers often represent the difference or translation between two points on the Argand diagram. For two complex numbers, say \(z_1\) and \(z_2\), the vector \(\overrightarrow{z_1 z_2}\) is found by subtracting \(z_1\) from \(z_2\). This is analogous to finding the difference in coordinates in a Cartesian plane.
For example, \[ \overrightarrow{AB} = B - A = (-1 + \mathrm{j}4) - (3 + \mathrm{j}2) = -4 + \mathrm{j}2 \] This result indicates that to move from A to B, we translate 4 units left and 2 units up, following the direction of the vector.
Vectors can be rotated by multiplying by the imaginary unit, \(\mathrm{j}\), representing a 90-degree rotation counterclockwise in the Argand plane. This is particularly useful when analyzing shapes like squares where sides are perpendicular.

The magnitude of a complex number is akin to finding the length of a vector from the origin to the point in the Argand diagram. It's a measure of how far the point is from the origin, much like the hypotenuse in a right triangle.
For a complex number \(z = a + \mathrm{j}b\), the magnitude is given by \[ |z| = \sqrt{a^2 + b^2} \] This formula arises from the Pythagorean theorem.
In the example, the magnitude of the vector \( \overrightarrow{AB} \) is calculated as\[ |\overrightarrow{AB}| = \sqrt{(-4)^2 + (2)^2} = \sqrt{16 + 4} = 2\sqrt{5} \] This result represents the physical distance between points A and B on the Argand plane.

Vectors are perpendicular when they intersect at a 90-degree angle, and the dot product of two perpendicular vectors is zero. This property is essential when determining whether four points form a square in an Argand diagram, as adjacent sides must be perpendicular.
Given two vectors \(\overrightarrow{u} = a + \mathrm{j}b\) and \(\overrightarrow{v} = c + \mathrm{j}d\), their dot product is calculated as \[ a \cdot c + b \cdot d \] If this result is zero, the vectors are perpendicular.
In our solution, the vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\) were shown to be perpendicular:\[ (-4)\times(-2) + 2\times(-4) = 8 - 8 = 0 \] Verifying this condition ensures the integrity of geometric shapes, reinforcing the properties of squares and rectangles illustrated in the Argand diagram.