Problem 87

Question

If for the complex numbers $z_{1}$ and $z_{2},\left|z_{1}+z_{2}\right|=\left|z_{1}-z_{2}\right|$, then $\operatorname{amp} z_{1} \sim \operatorname{amp} z_{2}=$ (A) $\pi$ (B) $\frac{\pi}{2}$ (C) $\frac{\pi}{4}$ (D) None of these

Step-by-Step Solution

Verified

Answer

(B) $\frac{\pi}{2}$

1Step 1: Understand the Problem Statement

We need to find the angle between the arguments (or amplitudes) of the complex numbers $z_1$ and $z_2$ given that $|z_1 + z_2| = |z_1 - z_2|$.

2Step 2: Analyze Vector Representation

The condition $|z_1 + z_2| = |z_1 - z_2|$ geometrically implies that the sum and difference of $z_1$ and $z_2$ have the same magnitude. This condition holds true if $z_1$ and $z_2$ are perpendicular because the magnitudes of the sums and differences of perpendicular vectors (or complex numbers) are equal.

3Step 3: Convert Problem to Trigonometric Form

If $z_1$ and $z_2$ are perpendicular, then the angle between them is $\frac{\pi}{2}$. The problem asks for the angle between their arguments, so $\operatorname{amp} z_1 \sim \operatorname{amp} z_2 = \frac{\pi}{2}.$

4Step 4: Verify Solution

Given $|z_1 + z_2| = |z_1 - z_2|$, the problem simplifies upon vector analysis or comparing it with trigonometric identities involving perpendicular angles to confirm that the angle between $z_1$ and $z_2$ is indeed $\frac{\pi}{2}.$

Key Concepts

Vector RepresentationAngle Between VectorsTrigonometric Form

Vector Representation

In the world of complex numbers, we can think of each number as a point or a vector in a two-dimensional plane. This makes complex numbers very handy for geometric interpretations. The complex number $ z = a + bi $ can be represented as a vector from the origin $(0,0)$ to the point $(a,b)$ in this plane, where $a$ is the real part and $b$ is the imaginary part.

When visualizing complex numbers as vectors, you can easily compare their directions and magnitudes, which helps in analyzing their relationships geometrically.
The magnitude (or absolute value) $|z|$ of a complex number $z$ is the length of the vector, calculated using the formula $|z| = \sqrt{a^2 + b^2}$.

In problems like the one we are discussing, understanding the vector representation helps us see that $|z_1 + z_2| = |z_1 - z_2|$ implies a geometric relationship between two vectors, suggesting they are perpendicular.

Angle Between Vectors

The angle between two vectors (or complex numbers) tells us how these vectors are oriented with respect to each other. A crucial part of understanding vector relationships involves analyzing the angle between them. The exercise problem, $|z_1 + z_2| = |z_1 - z_2|$, indicates a special relationship where the vectors are perpendicular.

When two vectors are perpendicular, the angle between them is $\frac{\pi}{2}$, which means they meet at a right angle.
In the context of complex numbers, this is an important geometric property since it simplifies to the relationship of their amplitudes or arguments.

By converting this relationship into an angle, students can better visualize how these complex numbers interact, leading us to find that $\operatorname{amp} z_1$ and $\operatorname{amp} z_2$ have an angle of $\frac{\pi}{2}$.

Trigonometric Form

The trigonometric form of a complex number provides a powerful tool for understanding their properties and relationships, especially useful in calculating angles and magnitudes. A complex number $z = a + bi$ in trigonometric form is represented as $r (\cos \theta + i \sin \theta)$, where $r$ is the magnitude $|z|$ and $\theta$ is the argument (angle from the positive real axis).

The trigonometric form is particularly useful when multiplying, dividing, or finding powers and roots of complex numbers because angles add up during multiplication.
In the problem you're facing, using trigonometric form simplifies comparison of the angles, confirming that the angle between $z_1$ and $z_2$ indeed simplifies to a right angle ($\frac{\pi}{2}$ radians).

Understanding how to transition between Cartesian and trigonometric forms is key to solving complex number problems, making this form an invaluable tool for students learning about complex numbers.

Problem 86

Problem 88

Other exercises in this chapter

Problem 85

The integral solution of the equation $(1-i)^{n}=2^{n}$ is (A) $n=0$ (B) $n=1$ (C) $n=-1$ (D) None of these

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Problem 86

The locus of the complex number $z$ in an argand plane satisfying the equation $$ \operatorname{Arg}(z+i)-\operatorname{Arg}(z-i)=\frac{\pi}{2} \text { is } $

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Problem 88

The locus of the complex number $z$ in an argand plane satisfying the inequality $\log _{1 / 2}\left(\frac{|z-1|+4}{3|z-1|-2}\right)>1\left(\right.$ where \

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Problem 89

Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{1}}=1$, then (A) $z_{1}, z_{2}$ are collinear (B) \(z_{1},

View solution