Why does rotating by $\frac{\pi}{2}$ mean multiplying by $i$?

This is a fundamental property of complex numbers. Any rotation by an angle $\theta$ about the origin is achieved by multiplying by $e^{i\theta}$. For $\theta = \frac{\pi}{2}$, we use Euler's formula: $e^{i\pi/2} = \cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2})$. Since $\cos(\frac{\pi}{2}) = 0$ and $\sin(\frac{\pi}{2}) = 1$, this simplifies to $0 + i(1) = i$.

JEE Advanced 2008

LEVELJEE Main

Animated Solution for Mathematics - Complex Numbers: A particle $P$ starts from the point $z_{0} = 1 + 2 i$ , where $i = - 1$ . It moves horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z_{1}$ . From $z_{1}$ the particle moves $2$ units in the direction of the vector $\hat{i} + \hat{j}$ and then it moves through an angle $\frac{π}{2}$ in anticlockwise direction on a circle with centre at origin, to reach a point $z_{2}$ . The point $z_{2}$ is given by

Q: Why do we add $(5 + 3i)$ to $z_0$?

In the Argand plane, a complex number $z = x + iy$ represents a point $(x, y)$. A horizontal shift of 5 units means the real part increases by 5, and a vertical shift of 3 units means the imaginary part increases by 3. Thus, the displacement is $5 + 3i$. Adding this to the initial position $z_0$ gives the new position $z_1$.

Q: How do we know the displacement vector is $1 + i$?

The problem states the particle moves $\sqrt{2}$ units in the direction of $\hat{i} + \hat{j}$. The vector $\hat{i} + \hat{j}$ corresponds to the complex number $1 + i$. The magnitude of $1 + i$ is $\sqrt{1^2 + 1^2} = \sqrt{2}$. Since the required displacement magnitude is $\sqrt{2}$ and the direction is the same, the displacement is exactly $1 + i$.

Select Answer:

Progressive Hints

Hint 1

Visualize the Argand plane as a map. The particle's journey is a sequence of vector additions followed by a rotation. Think of the complex plane not just as numbers, but as a 2D space where horizontal movement affects the real part and vertical movement affects the imaginary part.

Hint 2

Remember that in the complex plane, a horizontal shift of

a

and a vertical shift of

b

is equivalent to adding the complex number

a + bi

. For the rotation, recall that rotating a complex number

z

by an angle

θ

about the origin is equivalent to multiplying

z

e^{i θ}

Hint 3

Start by calculating

z_{1} = z_{0} + (5 + 3 i)

. Then, identify the displacement vector for the second move. Since it moves

2

units in the direction of

\hat{i} + \hat{j}

, the displacement is

1 + i

. Calculate

z_{1}^{'} = z_{1} + (1 + i)

. Finally, apply the rotation operator

e^{iπ /2} = i

z_{1}^{'}

to find

z_{2}

Visualized Solution

Initial Position $z_{0}$

Initial position: $z_{0} = 1 + 2 i$
Plotted as $(1, 2)$ on the Argand Plane

First Displacement Logic

Horizontal movement $\to$ Change in Real part
Vertical movement $\to$ Change in Imaginary part

Calculating $z_{1}$

$z_{1} = z_{0} + (Δ x + i Δ y)$
$z_{1} = (1 + 2 i) + (5 + 3 i)$

Position $z_{1}$

$z_{1} = (1 + 5) + (2 + 3) i$
$z_{1} = 6 + 5 i$

Vector Displacement

Moves $2$ units along $\hat{i} + \hat{j}$
Direction vector: $v = 1 + i$

Displacement Complex Number

Magnitude of $1 + i = 1^{2} + 1^{2} = 2$
Required displacement is exactly $1 + i$

Intermediate Point $z_{1}^{'}$

$z_{1}^{'} = z_{1} + (1 + i)$
$z_{1}^{'} = (6 + 5 i) + (1 + i)$

Calculating $z_{1}^{'}$

$z_{1}^{'} = (6 + 1) + (5 + 1) i$
$z_{1}^{'} = 7 + 6 i$

Rotation Concept

Rotation by $\frac{π}{2}$ anticlockwise about origin
Equivalent to multiplying by $e^{iπ /2}$

Rotation Operator

Euler's formula: $e^{iπ /2} = cos (\frac{π}{2}) + i sin (\frac{π}{2})$
$e^{iπ /2} = 0 + i (1) = i$

Setting up $z_{2}$

$z_{2} = z_{1}^{'} \times i$
$z_{2} = (7 + 6 i) \times i$

Distributing $i$

$z_{2} = 7 (i) + 6 i (i)$
$z_{2} = 7 i + 6 i^{2}$

Final Calculation

Recall: $i^{2} = - 1$
$z_{2} = 7 i + 6 (- 1)$
$z_{2} = - 6 + 7 i$

Conclusion

Final Position: $z_{2} = - 6 + 7 i$
Matches Option 4

00:00 / 00:00

The Sigma Insight: Geometrical Applications of Complex Numbers

Solution Diagram

Analyzing the Initial Translation

We begin our journey on the Argand plane at the starting coordinate

z_{0} = 1 + 2 i

The first movement is a translation of 5 units horizontally and 3 units vertically, which corresponds to adding the complex number

5 + 3 i

We calculate the new position

z_{1}

as follows:

z_{1} = (1 + 2 i) + (5 + 3 i) = 6 + 5 i

The Diagonal Leap

Next, the particle moves

2

units in the direction of the vector

\hat{i} + \hat{j}

. This vector is represented by the complex number

1 + i

We verify the magnitude of this displacement:

∣1 + i ∣ = 1^{2} + 1^{2} = 2

Since the magnitude matches our required displacement, we add this vector to our current position

z_{1}

to find the intermediate point

z_{1}^{'}

z_{1}^{'} = (6 + 5 i) + (1 + i) = 7 + 6 i

The Grand Rotation

To perform the final transformation, we rotate the point

z_{1}^{'}

\frac{π}{2}

radians in the anticlockwise direction about the origin. In the complex plane, this rotation is achieved by multiplying by

e^{i θ}

, where

θ = \frac{π}{2}

Using Euler's formula, we determine the rotation operator:

e^{iπ /2} = cos (\frac{π}{2}) + i sin (\frac{π}{2}) = 0 + i (1) = i

We now multiply our intermediate position

z_{1}^{'}

i

to find the final destination

z_{2}

z_{2} = (7 + 6 i) \times i

z_{2} = 7 i + 6 i^{2}

Given that

i^{2} = - 1

, we simplify the expression:

z_{2} = 7 i + 6 (- 1) = - 6 + 7 i

The final position of the particle is $- 6 + 7 i$ .

Common Traps & FAQs

Why do we add

(5 + 3 i)

z_{0}

In the Argand plane, a complex number

z = x + i y

represents a point

(x, y)

. A horizontal shift of 5 units means the real part increases by 5, and a vertical shift of 3 units means the imaginary part increases by 3. Thus, the displacement is

5 + 3 i

. Adding this to the initial position

z_{0}

gives the new position

z_{1}

How do we know the displacement vector is

1 + i

The problem states the particle moves

2

units in the direction of

\hat{i} + \hat{j}

. The vector

\hat{i} + \hat{j}

corresponds to the complex number

1 + i

. The magnitude of

1 + i

1^{2} + 1^{2} = 2

. Since the required displacement magnitude is

2

and the direction is the same, the displacement is exactly

1 + i

Why does rotating by

\frac{π}{2}

mean multiplying by

i

This is a fundamental property of complex numbers. Any rotation by an angle

θ

about the origin is achieved by multiplying by

e^{i θ}

. For

θ = \frac{π}{2}

, we use Euler's formula:

e^{iπ /2} = cos (\frac{π}{2}) + i sin (\frac{π}{2})

. Since

cos (\frac{π}{2}) = 0

and

sin (\frac{π}{2}) = 1

, this simplifies to

0 + i (1) = i

Select Answer:

Visualized Solution

Initial Position $z_{0}$

First Displacement Logic

Calculating $z_{1}$

Position $z_{1}$

Vector Displacement

Displacement Complex Number

Intermediate Point $z_{1}^{'}$

Calculating $z_{1}^{'}$

Rotation Concept

Rotation Operator

Setting up $z_{2}$

Distributing $i$

Final Calculation

Conclusion

The Sigma Insight: Geometrical Applications of Complex Numbers

Analyzing the Initial Translation

The Diagonal Leap

The Grand Rotation

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Select Answer:

Visualized Solution

Initial Position .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z0​

First Displacement Logic

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z1​

Position .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z1​

Vector Displacement

Displacement Complex Number

Intermediate Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z1′​

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z1′​

Rotation Concept

Rotation Operator

Setting up .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z2​

Distributing .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } i

Final Calculation

Conclusion

The Sigma Insight: Geometrical Applications of Complex Numbers

Analyzing the Initial Translation

The Diagonal Leap

The Grand Rotation

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