If z1=1+i, z2=3 i, z3=√(2)-i, then find (i) (z1 z̅2)/(z̅3)+(z2 z̅3)/(z1) (ii) z1 z2 z3+z1 z2 z3

If z1=1+i, z2=3 i, z3=√(2)-i, then find (i) (z1...

Key Concepts

Complex Numbers

Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit satisfying i² = -1. They extend the concept of one-dimensional real numbers to the two-dimensional complex plane, which makes possible the solution of equations that have no real solutions, and they are essential in various fields of mathematics and engineering.

Complex Conjugation

The complex conjugate of a complex number a + bi is given by a - bi. Complex conjugation is crucial because it reflects a number across the real axis in the complex plane and is used in simplifying expressions, particularly in division of complex numbers and in computing the modulus; it also has important properties such as (zw)¯ = z¯w¯ for any complex numbers z and w.

Modulus (Absolute Value) of a Complex Number

The modulus of a complex number, denoted |z|, is the distance from the origin to the point (a, b) in the complex plane, calculated as ?(a² + b²) for a complex number z = a + bi. It is used to measure the size of a complex number and appears in many formulas, including expressing a complex number in polar form and in properties like |zw| = |z|·|w|.

Arithmetic Operations with Complex Numbers

Arithmetic operations with complex numbers include addition, subtraction, multiplication, and division, which are performed by applying the distributive, associative, and commutative properties while taking into account that i² = -1. These operations are fundamental in solving problems involving complex expressions, as they allow one to combine and simplify complex numbers, and are used in a variety of applications across mathematics and physics.

Transcript

00:01 In the given question we are told that if z1 if the complex number z1 is equal to 1 plus i and the complex number z2 is equal to 3i and the complex number z 3 is given as root 2 minus i what is the following results so the first expression that we need to find that we need to evaluate is z1 times z2 bar z2 bar is denoting the conjugate of the complex number z2 divided by z 3 bar plus z 2 times z 3 bar divided by z z 3 bar divided by z 1 right so let's substitute the complex number in the in the given so what we will have is 1 plus i times the conjugate of z2 that is the z2 is equal to 3i so z 2 bar would be equal to z 2 bar is equal to minus 3i right so we will have 1 plus i times minus 3 i divided by z 3 bar so z 3 bar is the conjugate of the complex number z 3 which would be given by root 2 plus i.

01:33 So let's make this substitution.

01:36 So root 2 plus i plus z2 which is 3 i times z 3 bar which is again root 2 plus i and this is divided by z1 which is given as 1 plus i.

01:55 So this is what we have right so we can simplify the numerators so what we will have is minus 3i plus 3 minus 3 i square minus 3 i minus 3 i squared divided by root 2 plus i plus we have 3 i times root 2 which is 3 root 2 i plus 3 i times i which is 3 i squared divided by 1 plus i so what we will have is 3 i is minus 3 i minus i is equal to minus 1.

02:35 So minus 3 times minus 1 is plus 3.

02:39 So 3 minus 3 i divided by root 2 plus i plus 3 i square is 3 root 2 i or i square is minus 1 so minus 3 plus 3 root 2 i and this is divided by 1 plus i right so this is what we have after simplifying the complex numbers so now next what we should do is to take the simplify the denominators of each fraction by taking the product of 3 minus 3i times root 2 minus i in the numerator and the denominator in the first fraction root 2 minus i and in the second fraction we can take minus 3 plus 3 root 2 i times 1 minus i.

03:40 So 1 plus i times 1 minus i also.

03:44 So what we will have after multiplying this is we would get the result as 1 plus i we can take this as we just simplify the numerator.

04:03 2 minus 3 i minus 3 root 2 i minus 3 root 2 i minus plus 3 i square which would be plus 3 i square is equal to minus 1 so it is equal to 3 divided by root 2 plus 1 times root 2 minus 1 is root 2 square minus i square so it will be equal to 2 plus 1 which is equal to 3 and in the second fraction we will have minus 3 minus 3 i plus 3 i plus 3 root 2 i minus 2 i squared divided by 1 plus i times 1 minus i is of the form a plus b times a minus b which we can take as a square minus b square so over here it would be 1 square minus i square.

05:05 I square is equal to minus 1 so 1 plus 1 would give us 2 so now we can further simplify this as 3 root 2 minus 3 root 2 plus minus i 3 root 2 times 3 root 2 plus 3 right divided by 3 and in the second fraction what we have is minus 3 3.

05:38 I square is equal to minus 1 so plus 3 root 2 plus i times 3 plus 3 root 2 divided by 2 so let's take this as this is one term this is the real part over here in the fractions so now on cross multiplying what we can have what we will get is 2 times 3 root 2 minus 3 minus 2 times i times 3 root 2 plus 3 plus we can take 3 times minus 3 plus 3 root 2 plus 3 times 3 plus 3 root 2 plus 3 times 3 plus 3 root 2 divided by 3 times 2 which is equal to 6.

06:44 So over here we can see if we take a minus outside from the first term in the fraction which is this term what we get as minus 2 times no we don't have to take a minus outside right it is the same term 3 root 2 minus 3 and minus 3 plus 3 root 2 is the same term so we can just add them so what we will get is 5 times 3 root 2 minus 3 in the real part and in the imaginary part we can add 3i times 3 plus 3 root 2 minus 2 i times 3 root 2 plus 3 so we will have plus i 3 plus i 3 plus 3 root 2 and this is divided by 6 so then what we get as the imaginary as the complex number is 5 by 6 times 3 root 2 minus 3 in the real part plus i times 1 by 6 times 3 plus 3 root 2.

08:01 So we can take 3 as a common factor outside and 3 times 2 is 6 so what we will have is 5 by 2 times root 2 minus 1 as the real part and i times 1 by 2 times 1 plus root 2 as the imaginary part.

08:24 So this is the final answer that we get after simplifying the expression that is given in the question.

08:33 Now next let's look at the second expression that is given in the question which we can take as z 1 times z 2 bar, z 1 times z 2 bar times z 3 bar plus z 1 bar times z 2 times z 3 and this we can write as we already know z 1, z 2 and z 3 so we can write this as let's take a 2, each of this and what we will have is z1 is given as 1 plus i, z2 is given as 3i, and z3 is given as root 2 minus i.

09:37 So these are the complex numbers.

09:40 So now what we can do is we can take these complex numbers.

09:46 And multiply them...

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