(a) Prove that the complex numbers z and w correspond to...

(a) Prove that the complex numbers $z$ and $w$ correspond to orthogonal vectors in $\mathbb{R}^2$ if and only if $\operatorname{Re} z \bar{w}=0$.
(b) Prove that $z$ and i $z$ are always orthogonal.

Key Concepts

Rotation in the Complex Plane by Multiplication with i

Multiplying a complex number by i results in a 90° counterclockwise rotation in the complex plane. This rotation property is fundamental because it guarantees that any complex number and its image under multiplication by i are perpendicular. Therefore, the concept of rotation by i provides a straightforward geometric interpretation of orthogonality between z and i*z.

Complex Numbers as Vectors

Complex numbers can be represented as points or vectors in the plane, where a complex number z = a + bi corresponds to the vector (a, b) in ?². This geometric interpretation bridges the gap between algebra and geometry, allowing us to apply vector concepts such as length, direction, and orthogonality to complex numbers.

Dot Product

The dot product is an operation that takes two vectors and returns a scalar value, representing the product of their magnitudes and the cosine of the angle between them. It is a critical concept in determining orthogonality; two vectors are orthogonal if and only if their dot product equals zero. When complex numbers are viewed as vectors, this concept can be expressed in terms of their real and imaginary parts.

Complex Conjugation and the Real Part Operation

The complex conjugate transforms a complex number z = a + bi into ? = a - bi. The expression Re(z * ?) is significant because it gives the standard dot product of the vectors corresponding to z and w. This equivalence allows one to analyze orthogonality using complex multiplication and the real part function, instead of directly computing the dot product of the real components.

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