What point in the complex plane represents \(4-3i\)?

State the vector rule for adding \(z_1=a_1+b_1i\) and \(z_2=a_2+b_2i\).

Find the vector represented by \((2+i)+(3+4i)\).

Which complex number represents the vector from the origin to \((-1,6)\)?

Add \(-2+5i\) and \(7-3i\), and describe the geometric operation.

Find \((1-4i)-(6+i)\) and interpret it as a displacement.

If \(z_1=3+2i\) and \(z_1+z_2=-1+5i\), find \(z_2\).

Find the fourth vertex of the parallelogram with adjacent vertices \(0\), \(2+i\), and \(-1+4i\).

Let \(z_1=4+i\) and \(z_2=-2+3i\). Verify the triangle inequality for this pair.

Find the midpoint of the segment joining \(3-5i\) and \(-1+i\) in complex form.

Explain why \(z_1-z_2\) represents the vector from \(z_2\) to \(z_1\).

A student says \(|z_1+z_2|=|z_1|+|z_2|\) for all complex numbers. Give a counterexample and explain it geometrically.

The points \(0\), \(z\), \(w\), and \(z+w\) form a parallelogram. For \(z=2-3i\) and \(w=-5+i\), find both diagonals as complex displacements.

Find \(z\) if the parallelogram with vertices \(0\), \(2+i\), \(z\), and \(5-2i\) has \(5-2i\) as the vertex opposite \(0\).

For which real \(t\) are \(2+ti\) and \(-4-2ti\) opposite vectors?

For which real \(t\) is the sum \((t+2i)+(3-ti)\) purely real?

For which real \(t\) is the displacement from \(1+ti\) to \(4-2i\) horizontal?

Prove the parallelogram identity \(|z+w|^2+|z-w|^2=2|z|^2+2|w|^2\) for \(z=a+bi\), \(w=c+di\).

Show that equality in \(|z+w|\le |z|+|w|\) holds for \(z=2(1+i)\) and \(w=5(1+i)\), and explain why.

A learner claims that if \(|z+w|=|z-w|\), then \(z=0\) or \(w=0\). Disprove the claim geometrically with a complex example.