Animated Solution for Mathematics - Complex Numbers: A man walks a distance of 3 units from the origin towards the north-east ($N45^{\circ }E$) direction. From there, he walks a distance of 4 units towards the north-west ($N45^{\circ }W$) direction to reach a point $P$. Then the position of $P$ in the Argand plane is

A Journey Across the Plane

Imagine you are standing at the origin of a vast, infinite map. In the world of mathematics, we call this the Argand plane. It is not just a flat surface; it is a playground where algebra meets geometry.

Today, we are going to trace the path of a traveler moving through this plane. We will discover how complex numbers turn a seemingly tedious vector addition problem into a beautiful, elegant dance of rotations.

Before we move, we must define our coordinate system. In the Argand plane, the positive Real axis is our 'East', and the positive Imaginary axis is our 'North'. Our traveler starts at the origin $O(0,0)$.

Every step they take can be represented as a complex number $z=r{e}^{i\theta }$, where $r$ is the distance covered and $\theta$ is the angle measured counter-clockwise from the positive Real axis. This polar form is our most powerful tool; it allows us to encode both magnitude and direction into a single, compact expression.

Our traveler heads North-East. North-East is exactly halfway between North ($90^{\circ }$) and East ($0^{\circ }$), giving us an angle of ${\theta }_1=45^{\circ }$, or $\frac{\pi }{4}$ radians.

With a distance of $r_1=3$, the first displacement is simply:

This is the starting point of our vector chain. It is simple, clean, and perfectly defined.

Now, the traveler turns towards the North-West. North-West lies between North ($90^{\circ }$) and West ($180^{\circ }$). This puts our direction at exactly $135^{\circ }$, or $\frac{3\pi }{4}$ radians.

The traveler walks a distance of $r_2=4$. Thus, the second displacement is:

To find the final position $P$, we simply add these two displacements: $z_P=z_1+z_2$. This gives us:

At first glance, this looks like two separate terms that cannot be combined. But look closer! Mathematics loves symmetry. Notice that $\frac{3\pi }{4}$ is just $\frac{\pi }{4}+\frac{\pi }{2}$.

Using the laws of exponents, we can rewrite the second term:

Here is where the beauty of complex numbers shines. Euler's identity tells us that $e^{i\frac{\pi }{2}}=\cos (\frac{\pi }{2})+i\sin (\frac{\pi }{2})$. Since $\cos (90^{\circ })=0$ and $\sin (90^{\circ })=1$, this expression simplifies perfectly to $i$.

This is the 'rotation operator'—multiplying by $i$ is equivalent to a $90^{\circ }$ counter-clockwise rotation in the plane. Substituting this back into our equation, we get:

Now, the path clears. We can factor out the common term $e^{i\frac{\pi }{4}}$:

And there it is! We have arrived at the final position $P$. By treating the movements as complex numbers, we avoided the messy trigonometry of adding components and instead used the inherent rotational properties of the complex plane.

Remember, whenever you see a change in direction, look for a way to factor out the rotation. You are not just solving a problem; you are mastering the language of the universe. Keep pushing, keep visualizing, and the math will always reveal its hidden symmetry.