Problem 157 - A Complete Resource Book in Mathematics for JEE Main

Question

\(z_{1}=a+i b\) and \(z_{2}=c+i d\) are complex numbers such that \(\left|z_{1}\right|=\left|z_{2}\right|=1\) and \(\operatorname{Re}\left(z_{1} \bar{z}_{2}\right)=0 .\) If \(w_{1}=a+i c\) and \(w_{2}=b+i d(a, b, c, d \in R)\), then (A) \(\left|w_{1}\right|=1\) (B) \(\left|w_{2}\right|=1\) (C) \(\operatorname{Re}\left(w_{1} \bar{w}_{2}\right)=0\) (D) \(\operatorname{Re}\left(w_{1} \bar{w}_{2}\right)=1\)

Step-by-Step Solution

Answer

The correct answer is (B): \(|w_2| = 1\).

Step 1: Understanding Given Conditions

We have two complex numbers \(z_1 = a + ib\) and \(z_2 = c + id\). It is given that \(|z_1| = |z_2| = 1\), which implies \(a^2 + b^2 = 1\) and \(c^2 + d^2 = 1\). Additionally, \(\operatorname{Re}(z_1\bar{z}_2) = 0\) is given, meaning the dot product of their polar representations is zero.

Step 2: Analyzing the Dot Product Condition

The condition \(\operatorname{Re}(z_1\bar{z}_2) = ac + bd = 0\) implies a purely imaginary product, indicating that vectors representing \(z_1\) and \(z_2\) are perpendicular.

Step 3: Exploring Option A: \(|w_1| = 1\)

Define \(w_1 = a + ic\). Calculate \(|w_1|\): \(|w_1| = \sqrt{a^2 + c^2}\). Given \(a^2 + b^2 = 1\) and \(c^2 + d^2 = 1\), \(a^2 + c^2 = 1 - b^2 + 1 - d^2 - (a^2 + c^2)\). Verify this does not simplify to 1 always.

Step 4: Exploring Option B: \(|w_2| = 1\)

Define \(w_2 = b + id\). To find the magnitude \(|w_2|\), calculate \(|w_2| = \sqrt{b^2 + d^2}\). Since \(b^2 + a^2 = 1\) and \(c^2 + d^2 = 1\), that implies \(b^2 + d^2 = 1\), so \(|w_2| = 1\). Option B is correct.

Step 5: Exploring Option C: \(\operatorname{Re}(w_1 \bar{w}_2) = 0\)

Calculate \(w_1 \bar{w}_2 = (a + ic)(b - id) = ab + cd + i(cb - ad)\). The real part should be zero, \(ab + cd = 0\). With \(ac + bd = 0\), manipulate to find \(ab + cd = 0\) if and only if \(-ab = cd\). These do not necessarily have the same solutions.

Step 6: Exploring Option D: \(\operatorname{Re}(w_1 \bar{w}_2) = 1\)

From the expression \(ab + cd = 0\), it is unlikely this will equal to 1 unless further constraints are specified. With given data, Option D is incorrect.

Key Concepts

Magnitude of Complex Numbers

In the world of complex numbers, the magnitude is a measure of how long the vector associated with the complex number is. For a complex number \(z = a + ib\), its magnitude, also known as the modulus, is calculated using the formula \( |z| = \sqrt{a^2 + b^2} \).
Think of it as the distance from the origin on a complex plane to the point \((a, b)\). This is very similar to how you'd find the length of a diagonal in a right-angled triangle, using the Pythagorean theorem.

• Magnitude is always a non-negative real number.
• When the magnitude is 1, the complex number lies on the unit circle.

In our given problem, both \(|z_1| = 1\) and \(|z_2| = 1\), indicating they are on the unit circle. This implies that \(a^2 + b^2 = 1\) and \(c^2 + d^2 = 1\), important factors in understanding the constraints and exploring further solutions.

Perpendicular Vectors

In geometry, two vectors are said to be perpendicular if they meet at a right angle. For complex numbers, this concept translates into a condition on their real and imaginary parts when subjected to complex conjugation and multiplication.
If we have \(z_1 = a + ib\) and \(z_2 = c + id\), the product \(z_1 \bar{z}_2\) has a real part given by \(\operatorname{Re}(z_1 \bar{z}_2) = ac + bd\). If this real part equals zero, the vectors corresponding to these complex numbers are perpendicular.

• Perpendicularity implies that the dot product of their polar forms is zero.
• This ensures that the angle between the vectors is \(90^\circ\).

In our exercise, it's given that \(\operatorname{Re}(z_1 \bar{z}_2) = 0\), thus confirming \(ac + bd = 0\). This condition is pivotal when analyzing the options provided, especially in identifying perpendicular vector relations between \(z_1\) and \(z_2\).

Real Part of a Complex Number

The real part of a complex number is the component that doesn't involve the imaginary unit \(i\). For example, in a complex number \(z = a + ib\), the real part is \(a\). This is essential for understanding the nature of operations involving complex conjugates and products.
When multiplying two complex numbers, such as \(w_1 = a + ic\) and the conjugate of \(w_2 = b - id\), the form of their product is given by \(w_1 \bar{w}_2 = (a + ic)(b - id)\). Here, the real part becomes \(ab + cd\).

• The real part indicates the horizontal component in the complex plane.
• It plays a crucial role in determining orthogonal (perpendicular) vectors.

Analyzing whether \(\operatorname{Re}(w_1 \bar{w}_2) = 0\) or \(1\) is necessary to evaluate options C and D in the exercise. Our calculations show \(ab + cd = 0\) implying that the real parts suggest perpendicularity under certain conditions, similarly as in the original complex numbers.