Complex Numbers
Rotation – Multi-Step Geometric Movement
Complex Numbers_PYQ
Grade 11
Question:
A particle $P$ starts from the point $z_0 = 1+2i$, where $i=\sqrt{-1}$. It moves first 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 $\sqrt{2}$ units in the direction of the vector $\hat{i}+\hat{j}$ and then it moves through an angle $\dfrac{\pi}{2}$ in anti-clockwise direction on a circle with centre at origin, to reach a point $z_2$. The point $z_2$ is given by
$6+7i$
$-7+6i$
$7+6i$
$-6+7i$
Step-by-Step Solution
Key Concept: Anti-clockwise rotation by $\theta$ about the origin is multiplication by $e^{i\theta}$. Here $e^{i\pi/2}=i$, so $z_2=i(7+6i)=-6+7i$.
**Step 1: Find z₁ after horizontal and vertical moves**
$z_0=1+2i$. Moving horizontally away from origin: $\text{Re}(z_0)=1>0$, so move in $+x$ direction by $5$ units: $1+5+2i=6+2i$. Then moving vertically away from origin: $\text{Im}=2>0$, move in $+y$ direction by $3$ units: $z_1=6+5i$.
**Step 2: Move √2 units in direction î+ĵ**
Unit vector in direction $\hat{i}+\hat{j}$ is $\dfrac{1+i}{\sqrt{2}}$. Displacement $=\sqrt{2}\cdot\dfrac{1+i}{\sqrt{2}}=1+i$. New position: $6+5i+1+i=7+6i$.
**Step 3: Rotate by π/2 anti-clockwise about origin**
Multiply by $e^{i\pi/2}=i$: $z_2=(7+6i)\cdot i = 7i+6i^2 = -6+7i$.
Correct Answer: 4