What geometric effect does multiplication by \(i\) have in the complex plane?

State the modulus rule for a product \(z_1z_2\).

Find the image of \(3+2i\) after multiplication by \(-1\).

Find the image of \(4-i\) after multiplication by \(2\).

Multiply \(1+i\) by \(i\) and describe the rotation.

If \(z=2\operatorname{cis}(\pi/6)\), find the modulus and argument of \(3z\).

If \(z=5\operatorname{cis}(\pi/4)\), find the polar form of \(iz\).

Find the geometric effect of multiplying by \(2\operatorname{cis}(-\pi/3)\).

Use polar form to find \((1+i)(\sqrt3+i)\).

A point has complex coordinate \(2-2i\). Find its image after multiplication by \(\operatorname{cis}(\pi/2)\).

Explain why multiplication by a unit complex number preserves distance from the origin.

A student says multiplying by \(1+i\) is only a rotation by \(\pi/4\). What is missing?

Find the multiplier that maps \(1\) to \(-3i\), and describe its geometry.

Find the complex multiplier that rotates every point by \(\pi/6\) and halves its distance from the origin.

For which real \(t\) does multiplication by \(t+i\) preserve distances from the origin?

For which real \(t>0\) does multiplication by \(t\operatorname{cis}(\pi/3)\) triple areas in the plane?

For which real \(t\) is multiplication by \(2+ti\) a scaling by \(\sqrt5\)?

Prove that multiplication by \(re^{i\theta}\), with \(r>0\), scales lengths by \(r\) and rotates arguments by \(\theta\).

Show that multiplication by a non-zero complex number maps straight rays from the origin to straight rays from the origin.

A learner claims that multiplying by any complex number only rotates the plane. Disprove this and state the correct rule.