If the vertices of a square z1,z2,z3 and z4 taken in the anti-clockwise order, then z3=
- -iz1-(1+i)z2
- z1-(1+i)z2
- z1+(1+i)z2
- -iz1+(1+i)z2
The Correct Option is D
Solution and Explanation
The expression z3 = -iz1 + (1+i)z2 represents the location of vertex z3 of a square when the vertices are labeled in anti-clockwise order.
-iz1: This term represents a rotation of z1 by 90 degrees in the counterclockwise direction around the origin. Since the vertices are listed in anti-clockwise order, moving from z1 to z3 involves a 90-degree rotation from z1.
(1+i)z2: This term represents a translation of z2 by a vector (1+i), which corresponds to moving one unit in the positive real direction and one unit in the positive imaginary direction from z2. This translation from z2 also contributes to reaching the position of z3.
Combining the effects of the 90-degree rotation (-iz1) and the translation by (1+i)z2, we arrive at the correct position for z3 when the vertices are listed in anti-clockwise order for a square.
This answer satisfies the requirement that z3 should be diagonally opposite to z1, and it also accounts for the position of z2 in the anti-clockwise order of vertices.
The correct answer is option (D): -iz1+(1+i)z2
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Concepts Used:
Algebra of Complex Numbers
Algebra of complex numbers
1. Addition of two complex numbers:
Consider z1 and z2 are two complex numbers.
For example, z1 = 3+4i and z2 = 4+3i
Here a=3, b=4, c=4, d=3
∴z1+ z2 = (a+c)+(b+d)i
⇒z1 + z2 = (3+4)+(4+3)i
⇒z1 + z2 = 7+7i
Properties of addition of complex numbers
- Closure law: While adding two complex numbers the resulting number is also a complex number.
- Commutative law: For the complex numbers z1 and z2 , the commutation can be z1+ z2 = z2+z1
- Associative law: While considering three complex numbers, (z1+ z2) + z?3 = z1 + (z2 + z3)
- Additive identity: An Additive identity is nothing but zero complex numbers that go as 0+i0. For every complex number z, z+0 = z.
- Additive inverse: Every complex number has an additive inverse denoted as -z.
2. Difference between two complex numbers
It is similar to the addition of complex numbers, such that, z1 - z2 = z1 + ( -z2)
For example: (5+3i) - (2+1i) = (5-2) + (-2-1i) = 3 - 3i
3. Multiplication of complex numbers
Considering the same value of z1 and z2 , the product of the complex numbers are
z1 * z2 = (ac-bd) + (ad+bc) i
For example: (5+6i) (2+3i) = (5×2) + (6×3)i = 10+18i
Properties of Multiplication of complex numbers
Note: The properties of multiplication of complex numbers are similar to the properties we discussed in addition to complex numbers.
- Closure law: When two complex numbers are multiplied the result is also a complex number.
- Commutative law: z1* z2 = z2 * z1
Associative law: Considering three complex numbers, (z1 z2) z3 = z1 (z2 z3)
- Multiplicative identity: 1+0i is always denoted as 1. This is multiplicative identity. This means that z.1 = z for every complex number z.
- Distributive law: Considering three complex numbers, z1 (z2 + z3) =z1 z2 + z1 z3 and (z1+ z2) z3 = z1 z2 + z2 z3.
Read More: Complex Numbers and Quadratic Equations
4. Division of complex numbers
If z1 / z2 of a complex number is asked, simplify it as z1 (1/z2 )
For example: z1 = 4+2i and z2 = 2 - i
z1 / z2 =(4+2i)×1/(2 - i) = (4+i2)(2/(2²+(-1)² ) + i (-1)/(2²+(-1)² ))
=(4+i2) ((2+i)/5) = 1/5 [8+4i + 2(-1)+1] = 1/5 [8-2+1+41] = 1/5 [7+4i]
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