fin the product z1 * z2 and quotient z1/z2 using the standard form for z1 and z2.
z1 = 2 - 3i and z2 = 1 - sqrt 3i
Z1 = 2-3i, Z2 = 1-sqrt(3)i.
Z1*Z2 = (2-3i)(1-sqrt(3)i),
= 2-2sqrt(3)i-3i+3sqrt(3)*-1,
= 2-6.46i-5.2,
Z1*Z2 = -3.2 - 6.46i
Or
Convert both to polar form and get:
Z1*Z2 = (3.61,-56.3deg)(2,-60deg),
Z1*Z2=(7.22,-116.3deg) = -3.2 - 6.47i.
Z1/Z2 = (3.61,-56.3deg)/(2,-60deg),
Z1/Z2 = (1.8,3.7deg) = 1.796 - 0.1162i.
To find the product of z1 * z2, we can use the FOIL method:
Step 1: Multiply the real parts:
(2)(1) = 2
Step 2: Multiply the imaginary parts:
(-3i)(-sqrt(3)i) = 3sqrt(3)i^2 = 3sqrt(3)(-1) = -3sqrt(3)
Step 3: Combine the terms:
Product of z1 * z2 = 2 + (-3sqrt(3))i = 2 - 3sqrt(3)i
To find the quotient z1/z2, we multiply the numerator and denominator by the conjugate of the denominator:
Step 1: Multiply the numerator and denominator by the conjugate of z2:
z1 = 2 - 3i
z2 = 1 - sqrt(3)i
Conjugate of z2 = 1 + sqrt(3)i
Numerator: z1 * conjugate of z2
= (2 - 3i)(1 + sqrt(3)i)
= 2 + sqrt(3)i - 3i - 3sqrt(3)i^2
= 2 + sqrt(3)i - 3i - 3sqrt(3)(-1)
= 2 + sqrt(3)i - 3i + 3sqrt(3)
= 5 + sqrt(3)i
Denominator: z2 * conjugate of z2
= (1 - sqrt(3)i)(1 + sqrt(3)i)
= 1 + sqrt(3)i - sqrt(3)i - 3i^2
= 1 - 3i^2
= 1 - 3(-1)
= 1 + 3
= 4
Step 2: Divide the numerator by the denominator:
Quotient z1/z2 = (5 + sqrt(3)i) / 4
Therefore, the product of z1 * z2 is 2 - 3sqrt(3)i, and the quotient of z1/z2 is (5 + sqrt(3)i) / 4.
To find the product and quotient of complex numbers in standard form, we'll use the following formulas:
1. Product: (a + bi) * (c + di) = (ac - bd) + (ad + bc)i
2. Quotient: (a + bi) / (c + di) = [(ac + bd) + (bc - ad)i] / (c^2 + d^2)
Let's calculate the product first:
z1 = 2 - 3i
z2 = 1 - √3i
Product: z1 * z2
To find the real and imaginary parts of the product, we'll substitute the values in the formula:
(a + bi) * (c + di) = (ac - bd) + (ad + bc)i
Substituting the values:
(2 - 3i) * (1 - √3i) = (2 * 1 - (-3 * √3)) + (2 * √3 + (-3 * 1))i
= (2 + 3√3) + (2√3 - 3)i
Therefore, the product of z1 and z2 is:
z1 * z2 = (2 + 3√3) + (2√3 - 3)i
Let's move on to calculate the quotient:
Quotient: z1 / z2
To find the real and imaginary parts of the quotient, we'll substitute the values in the formula:
(a + bi) / (c + di) = [(ac + bd) + (bc - ad)i] / (c^2 + d^2)
Substituting the values:
(2 - 3i) / (1 - √3i) = [(2 * 1 + (-3 * -√3)) + (-3 * 1 - 2√3)i] / ((1)^2 + (√3)^2)
= (2 + 3√3) + (2√3 + 3)i / (1 + 3)
= (2 + 3√3) + (2√3 + 3)i / 4
Therefore, the quotient of z1 and z2 is:
z1 / z2 = [(2 + 3√3) + (2√3 + 3)i] / 4