FM - Core Pure 1 - 2) Argand diagrams

Question 1

Axes on an Argand diagram

Answer 1

• x-axis → real axis
• y-axis → imaginary axis

Question 2

Angles and directions on an Argand diagram

Answer 2

From positive to negative real axis:

• TR to TL → π^c (180°)
• BR to BL → -π^c (180°)

Question 3

Meaning and formula for the modulus of a complex number

Answer 3

• Distance from the origin to the complex, on an Argand diagram
• |z| = √[x² + y²]

Question 4

Meaning and formula for the argument of a complex number

Answer 4

• Angle between positive real axis and the line joining the complex number to the origin, on an Argand diagram
• tan^-1(y/x)

Question 5

Changes that need to be made to the argument (positive acute angle (α) made between the real axis and the modulus of a complex number) for each quadrant on an Argand diagram

Answer 5

• TR → arg(z) = α
• TL → arg(z) = π - α
• BL → arg(z) = a - π
• BR → arg(z) = -α

Question 6

Modulus-argument form of a complex number

Answer 6

z = r(cos(θ) + i·sin(θ))

Question 7

Manipulate |z₁z₂|

Answer 7

|z₁||z₂|

Question 8

Manipulate |z₁/z₂|

Answer 8

|z₁|/|z₂|

Question 9

Manipulate arg(z₁z₂)

Answer 9

arg(z₁) + arg(z₂)

Question 10

Manipulate arg(z₁/z₂)

Answer 10

arg(z₁) - arg(z₂)

Question 11

Meaning of |z₂ - z₁| on an Argand diagram

Answer 11

Distance between points z₁ and z₂

Question 12

Meaning of |z - z₁| = r or |z - (x + iy)| = r on an Argand diagram

Answer 12

Circle with centre (x, y) and radius r

Question 13

Meaning of |z - z₁| = |z - z₂| on an Argand diagram

Answer 13

Perpendicular bisector of the line segment joining z₁ and z₂