Let O be the origin, the point A be z1=3+22i, the point B(z2) be such that 3∣z2∣=∣z1∣ and arg(z2)=arg(z1)+2π. Then
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Step-by-Step Solution
Step 1: Calculate Modulus and Argument of z1
First, we need to find the modulus and argument of the complex number z1. The modulus of a complex number a+bi is given by a2+b2, and its argument is tan−1(b/a). For z1=3+22i, we calculate its modulus as (3)2+(22)2=3+8=11. The argument is tan−1(322).
Step 2: Calculate Modulus of z2
We are given the relation 3∣z2∣=∣z1∣. We can use the modulus of z1 we just calculated to find the modulus of z2. Dividing both sides by 3, we get ∣z2∣=311, which simplifies to 311.
Step 3: Calculate Argument of z2
We are given that the argument of z2 is 2π radians (or 90 degrees) greater than the argument of z1. This means that the vector representing z2 is obtained by rotating the vector representing z1 counter-clockwise by 90 degrees.
Step 4: Express z2 in terms of z1
We can express z2 in polar form using its modulus and argument. Since arg(z2)=arg(z1)+2π, we can write eiarg(z2)=eiarg(z1)ei2π. Also, we know that ei2π=cos(2π)+isin(2π)=0+i(1)=i. Substituting these into the polar form for z2, we get z2=3∣z1∣eiarg(z1)i. Since z1=∣z1∣eiarg(z1), we can write z2=3iz1.
Step 5: Substitute z1 to find z2
Now we substitute the given value of z1=3+22i into the expression for z2. We multiply 3i by z1. Remember that i2=−1. After distributing and simplifying, we get z2=−322+i.