Which Of The Following Represents Vector W = 11 I − 60 J W = 11i - 60j W = 11 I − 60 J In Trigonometric Form?A. W = 61\left(\sin 280.389^{\circ}, \cos 280.389^{\circ}\right ]B. W = 61\left(\cos 280.389^{\circ}, \sin 280.389^{\circ}\right ]C. $w =

Introduction

In mathematics, vectors can be represented in various forms, including trigonometric form. This form is particularly useful for expressing vectors in terms of their magnitude and direction. In this article, we will explore how to represent a given vector in trigonometric form and determine the correct representation among the given options.

Understanding Vectors

A vector is a mathematical object that has both magnitude (length) and direction. It can be represented graphically as an arrow in a coordinate system. The vector $w = 11i - 60j$ is a two-dimensional vector, where $i$ and $j$ are the unit vectors in the x and y directions, respectively.

Converting to Trigonometric Form

To convert a vector to trigonometric form, we need to find its magnitude and direction. The magnitude of a vector $w = xi + yj$ is given by:

$$|w| = \sqrt{x^2 + y^2}$$

In this case, the magnitude of the vector $w = 11i - 60j$ is:

$$|w| = \sqrt{11^2 + (-60)^2} = \sqrt{121 + 3600} = \sqrt{3721} = 61$$

The direction of the vector can be found using the inverse tangent function:

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

In this case, the direction of the vector $w = 11i - 60j$ is:

$$\theta = \tan^{-1}\left(\frac{-60}{11}\right) = \tan^{-1}(-5.4545) = -280.389^{\circ}$$

Representing the Vector in Trigonometric Form

Now that we have the magnitude and direction of the vector, we can represent it in trigonometric form. The trigonometric form of a vector $w = xi + yj$ is given by:

$$w = |w|(\cos \theta, \sin \theta)$$

In this case, the trigonometric form of the vector $w = 11i - 60j$ is:

$$w = 61(\cos (-280.389^{\circ}), \sin (-280.389^{\circ}))$$

Evaluating the Options

Now that we have the correct representation of the vector in trigonometric form, we can evaluate the given options.

A. $w = 61\left(\sin 280.389^{\circ}, \cos 280.389^{\circ}\right)$

B. $w = 61\left(\cos 280.389^{\circ}, \sin 280.389^{\circ}\right)$

C. $w = 61\left(\cos (-280.389^{\circ}), \sin (-280.389^{\circ})\right)$

The correct representation of the vector in trigonometric form is option C.

Conclusion

In this article, we explored how to represent a given vector in trigonometric form and determined the correct representation among the given options. We found that the vector $w = 11i - 60j$ can be represented in trigonometric form as $w = 61(\cos (-280.389^{\circ}), \sin (-280.389^{\circ}))$. This representation is useful for expressing vectors in terms of their magnitude and direction.

Key Takeaways

• Vectors can be represented in various forms, including trigonometric form.
• The trigonometric form of a vector is given by $w = |w|(\cos \theta, \sin \theta)$.
• The magnitude of a vector is given by $|w| = \sqrt{x^2 + y^2}$.
• The direction of a vector can be found using the inverse tangent function: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$.

Further Reading

For more information on vectors and trigonometric form, refer to the following resources:

• Vector Wikipedia Page
• Trigonometric Form of a Vector
• Vector Calculus
  Vector Representation in Trigonometric Form: Q&A ==============================================

Introduction

In our previous article, we explored how to represent a given vector in trigonometric form and determined the correct representation among the given options. In this article, we will answer some frequently asked questions related to vector representation in trigonometric form.

Q: What is the trigonometric form of a vector?

A: The trigonometric form of a vector is a way of representing a vector in terms of its magnitude and direction. It is given by:

$$w = |w|(\cos \theta, \sin \theta)$$

where $|w|$ is the magnitude of the vector and $\theta$ is the direction of the vector.

Q: How do I find the magnitude of a vector?

A: The magnitude of a vector is given by:

$$|w| = \sqrt{x^2 + y^2}$$

where $x$ and $y$ are the components of the vector.

Q: How do I find the direction of a vector?

A: The direction of a vector can be found using the inverse tangent function:

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

Q: What is the difference between the trigonometric form and the rectangular form of a vector?

A: The rectangular form of a vector is given by:

$$w = xi + yj$$

where $x$ and $y$ are the components of the vector. The trigonometric form of a vector is given by:

$$w = |w|(\cos \theta, \sin \theta)$$

where $|w|$ is the magnitude of the vector and $\theta$ is the direction of the vector.

Q: Can I convert a vector from rectangular form to trigonometric form?

A: Yes, you can convert a vector from rectangular form to trigonometric form using the following steps:

1. Find the magnitude of the vector using the formula:

   $$|w| = \sqrt{x^2 + y^2}$$

2. Find the direction of the vector using the inverse tangent function:

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

3. Represent the vector in trigonometric form:

   $$w = |w|(\cos \theta, \sin \theta)$$

Q: Can I convert a vector from trigonometric form to rectangular form?

A: Yes, you can convert a vector from trigonometric form to rectangular form using the following steps:

1. Find the magnitude of the vector:

   $$|w| = \sqrt{x^2 + y^2}$$

2. Find the direction of the vector:

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

3. Represent the vector in rectangular form:

   $$w = xi + yj$$

Q: What are some common applications of vector representation in trigonometric form?

A: Vector representation in trigonometric form has many applications in physics, engineering, and mathematics. Some common applications include:

• Representing forces and velocities in terms of their magnitude and direction
• Solving problems involving circular motion and rotation
• Analyzing and designing electrical circuits and electronic systems
• Modeling and simulating complex systems and phenomena

Conclusion

In this article, we answered some frequently asked questions related to vector representation in trigonometric form. We hope that this article has provided you with a better understanding of this important concept in mathematics and its applications in various fields.

Key Takeaways

• The trigonometric form of a vector is a way of representing a vector in terms of its magnitude and direction.
• The magnitude of a vector is given by $|w| = \sqrt{x^2 + y^2}$.
• The direction of a vector can be found using the inverse tangent function: $\theta = \tan^{-1}\left(\frac{y}{x}\right)$.
• Vector representation in trigonometric form has many applications in physics, engineering, and mathematics.

Further Reading

For more information on vector representation in trigonometric form, refer to the following resources:

• Vector Wikipedia Page
• Trigonometric Form of a Vector
• Vector Calculus