Vector interpretation of addition and conjugation

1. Find $u+v$ and $u-v$ for $u=3-2i$ and $v=-1+i$.

   3 marks
2. Find the vector from $A=-2+i$ to $B=1-3i$.

   2 marks
3. Find the midpoint between $2+5i$ and $-4-i$.

   3 marks
4. Describe the effect of multiplying all complex numbers by $-2$.

   3 marks
5. Prove $z+\overline{z}$ is real and $z-\overline{z}$ has real part zero for every complex $z$.

   4 marks
6. A parallelogram has successive vertices $0$, $z$, $z+w$, $w$. For $z=4+i$ and $w=-1+2i$, find its centre.

   3 marks

Equivalent correct methods earn the corresponding marks. Include units where asked.

1. Find $u+v$ and $u-v$ for $u=3-2i$ and $v=-1+i$.

   3 marks

   Add or subtract the two real components and the two imaginary components.

   $u+v=2-i$; $u-v=4-3i$.
2. Find the vector from $A=-2+i$ to $B=1-3i$.

   2 marks

   $B-A=\left(1-3i\right)-\left(-2+i\right)=3-4i$.

   $3-4i$.
3. Find the midpoint between $2+5i$ and $-4-i$.

   3 marks

   Average components: $\frac{2-4}{2}=-1$ and $\frac{5-1}{2}=2$.

   $-1+2i$.
4. Describe the effect of multiplying all complex numbers by $-2$.

   3 marks

   Both coordinates double in magnitude and reverse sign.

   A dilation by factor $2$ about the origin followed by a half-turn.
5. Prove $z+\overline{z}$ is real and $z-\overline{z}$ has real part zero for every complex $z$.

   4 marks

   Write the conjugate as $a-bi$.

   Add and subtract componentwise; each result has the required zero component.

   If $z=a+bi$, the expressions are $2a$ and $2bi$. The second is pure imaginary when $b\ne 0$, and is zero when $b=0$.
6. A parallelogram has successive vertices $0$, $z$, $z+w$, $w$. For $z=4+i$ and $w=-1+2i$, find its centre.

   3 marks

   The opposite vertices $0$ and $z+w=3+3i$ have midpoint $\frac{3+3i}{2}$.

   Diagonals of a parallelogram share their midpoint.

   $\frac{3}{2}+\frac{3}{2}i$.