Question

Let $z$ be a complex number. Then, the angle between vectors $z$ and $iz$ is

• A. $\pi$
• B. $0$
• C. $\frac{\pi }{2}$
• D. None of these

Answer 1

The correct option is C $\frac{\pi }{2}$

Let $z=x+iy$

Argument of $z$ $=\tan \theta =\frac{y}{x}$

Let $z^{\prime }=iz=i(x+iy)=-y+ix$

Argument of $z^{\prime }$ $=\tan {\theta }^{\prime }=-\frac{x}{y}$

Let the angle between the two complex numbers be $\alpha$

$\tan \alpha =\tan ({\theta }^{\prime }-\theta )=\frac{\tan {\theta }^{\prime }-\tan \theta }{1+\tan {\theta }^{\prime }\tan \theta }\Rightarrow \tan \alpha =\frac{\frac{y}{x}+\frac{x}{y}}{1-\frac{y}{x}\times \frac{x}{y}}=\infty \Rightarrow \alpha =\frac{\pi }{2}$

So option $C$ is correct.