Find The Direction Angle Of $3i + J$.A. 198.43 ∘ 198.43^{\circ} 198.4 3 ∘ B. 25.99 ∘ 25.99^{\circ} 25.9 9 ∘ C. 18.43 ∘ 18.43^{\circ} 18.4 3 ∘ D. 108.43 ∘ 108.43^{\circ} 108.4 3 ∘

Introduction

In mathematics, complex numbers are used to represent points in a two-dimensional plane. A complex number is typically denoted as $z = a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. The direction angle of a complex number is the angle it makes with the positive real axis in the complex plane. In this article, we will learn how to find the direction angle of a complex number.

What is a Complex Number?

A complex number is a number that can be expressed in the form $z = a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. The imaginary unit $i$ is defined as the square root of $-1$, denoted as $i = \sqrt{-1}$. Complex numbers can be represented graphically in the complex plane, which is a two-dimensional plane with the real axis and the imaginary axis.

The Complex Plane

The complex plane is a two-dimensional plane with the real axis and the imaginary axis. The real axis is the horizontal axis, and the imaginary axis is the vertical axis. Complex numbers are represented as points in the complex plane, with the real part of the complex number on the real axis and the imaginary part on the imaginary axis.

Finding the Direction Angle

The direction angle of a complex number is the angle it makes with the positive real axis in the complex plane. To find the direction angle of a complex number, we can use the following formula:

$$\theta = \tan^{-1}\left(\frac{b}{a}\right)$$

where $\theta$ is the direction angle, $a$ is the real part of the complex number, and $b$ is the imaginary part.

Example

Let's consider the complex number $3i + j$. To find the direction angle of this complex number, we can use the formula above.

$$\theta = \tan^{-1}\left(\frac{1}{3}\right)$$

Using a calculator, we can find the value of $\theta$.

$$\theta \approx 18.43^{\circ}$$

Therefore, the direction angle of the complex number $3i + j$ is approximately $18.43^{\circ}$.

Conclusion

In this article, we learned how to find the direction angle of a complex number. We used the formula $\theta = \tan^{-1}\left(\frac{b}{a}\right)$ to find the direction angle of the complex number $3i + j$. We also discussed the complex plane and how complex numbers are represented graphically in the complex plane. With this knowledge, you can now find the direction angle of any complex number.

Step-by-Step Solution

To find the direction angle of a complex number, follow these steps:

1. Write the complex number in the form $z = a + bi$.
2. Identify the real part $a$ and the imaginary part $b$ of the complex number.
3. Use the formula $\theta = \tan^{-1}\left(\frac{b}{a}\right)$ to find the direction angle.
4. Use a calculator to find the value of $\theta$.

Common Mistakes

When finding the direction angle of a complex number, make sure to:

• Use the correct formula: $\theta = \tan^{-1}\left(\frac{b}{a}\right)$.
• Identify the real part $a$ and the imaginary part $b$ of the complex number.
• Use a calculator to find the value of $\theta$.

Practice Problems

1. Find the direction angle of the complex number $2i + 3j$.
2. Find the direction angle of the complex number $4i - 2j$.
3. Find the direction angle of the complex number $5i + 6j$.

Answer Key

1. $\theta \approx 25.99^{\circ}$.
2. $\theta \approx 198.43^{\circ}$.
3. $\theta \approx 18.43^{\circ}$.

Final Answer

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Q: What is the direction angle of a complex number?

A: The direction angle of a complex number is the angle it makes with the positive real axis in the complex plane.

Q: How do I find the direction angle of a complex number?

A: To find the direction angle of a complex number, you can use the formula $\theta = \tan^{-1}\left(\frac{b}{a}\right)$, where $\theta$ is the direction angle, $a$ is the real part of the complex number, and $b$ is the imaginary part.

Q: What is the formula for finding the direction angle of a complex number?

A: The formula for finding the direction angle of a complex number is $\theta = \tan^{-1}\left(\frac{b}{a}\right)$.

Q: What is the imaginary unit $i$?

A: The imaginary unit $i$ is defined as the square root of $-1$, denoted as $i = \sqrt{-1}$.

Q: How do I represent a complex number graphically in the complex plane?

A: To represent a complex number graphically in the complex plane, you can plot the real part of the complex number on the real axis and the imaginary part on the imaginary axis.

Q: What is the complex plane?

A: The complex plane is a two-dimensional plane with the real axis and the imaginary axis.

Q: How do I find the real part and the imaginary part of a complex number?

A: To find the real part and the imaginary part of a complex number, you can write the complex number in the form $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Q: What is the difference between the real axis and the imaginary axis?

A: The real axis is the horizontal axis, and the imaginary axis is the vertical axis.

Q: Can I use a calculator to find the direction angle of a complex number?

A: Yes, you can use a calculator to find the direction angle of a complex number.

Q: What are some common mistakes to avoid when finding the direction angle of a complex number?

A: Some common mistakes to avoid when finding the direction angle of a complex number include:

• Using the wrong formula
• Identifying the real part and the imaginary part incorrectly
• Not using a calculator to find the value of $\theta$

Q: How do I practice finding the direction angle of a complex number?

A: You can practice finding the direction angle of a complex number by using the formula and working through examples.

Q: What are some examples of complex numbers?

A: Some examples of complex numbers include:

• $2i + 3j$
• $4i - 2j$
• $5i + 6j$

Q: Can I find the direction angle of a complex number with a negative real part?

A: Yes, you can find the direction angle of a complex number with a negative real part by using the formula and taking into account the sign of the real part.

Q: What is the final answer to the problem of finding the direction angle of the complex number $3i + j$?

A: The final answer to the problem of finding the direction angle of the complex number $3i + j$ is $\boxed{18.43^{\circ}}$.