Simplifying the Components

Oct 13, 2025 by windailys.com 140 views

Introduction

Vectors are mathematical objects that have both magnitude and direction. They are used to represent quantities with both size and direction, such as forces, velocities, and displacements. In this article, we will explore the concept of vector addition, which is a fundamental operation in vector mathematics. We will use the given vectors $u$ and $v$ to demonstrate the process of adding two vectors.

Vector $u$ and Vector $v$

The given vectors are:

Vector $u$ : $2\left\langle\cos 95^{\circ}, \sin 95^{\circ}\right\rangle$
Vector $v$ : $4\left\langle\cos 165^{\circ}, \sin 165^{\circ}\right\rangle$

To add these vectors, we need to first understand their components. The components of a vector are the values that make up the vector. In this case, the components of vector $u$ are $2\cos 95^{\circ}$ and $2\sin 95^{\circ}$ , while the components of vector $v$ are $4\cos 165^{\circ}$ and $4\sin 165^{\circ}$ .

Adding Vectors $u$ and $v$

To add vectors $u$ and $v$ , we need to add their corresponding components. This means that we add the x-components (the first components) and the y-components (the second components) separately.

Let's calculate the x-component of the sum $u + v$ :

\begin{aligned} \text{x-component of } u + v &= \text{x-component of } u + \text{x-component of } v \\ &= 2\cos 95^{\circ} + 4\cos 165^{\circ} \end{aligned}

Now, let's calculate the y-component of the sum $u + v$ :

\begin{aligned} \text{y-component of } u + v &= \text{y-component of } u + \text{y-component of } v \\ &= 2\sin 95^{\circ} + 4\sin 165^{\circ} \end{aligned}

Simplifying the Components

To simplify the components of the sum $u + v$ , we can use trigonometric identities. Specifically, we can use the angle addition formula for cosine and sine:

\begin{aligned} \cos (A + B) &= \cos A \cos B - \sin A \sin B \\ \sin (A + B) &= \sin A \cos B + \cos A \sin B \end{aligned}

Using these formulas, we can rewrite the x-component of the sum $u + v$ as:

\begin{aligned} \text{x-component of } u + v &= 2\cos 95^{\circ} + 4\cos 165^{\circ} \\ &= 2\cos (90^{\circ} + 5^{\circ}) + 4\cos (180^{\circ} - 15^{\circ}) \\ &= 2(\sin 5^{\circ}) + 4(-\cos 15^{\circ}) \end{aligned}

Similarly, we can rewrite the y-component of the sum $u + v$ as:

\begin{aligned} \text{y-component of } u + v &= 2\sin 95^{\circ} + 4\sin 165^{\circ} \\ &= 2\sin (90^{\circ} + 5^{\circ}) + 4\sin (180^{\circ} - 15^{\circ}) \\ &= 2(\cos 5^{\circ}) + 4(\sin 15^{\circ}) \end{aligned}

Evaluating the Components

Now that we have simplified the components of the sum $u + v$ , we can evaluate them using a calculator or a trigonometric table.

Let's evaluate the x-component of the sum $u + v$ :

\begin{aligned} \text{x-component of } u + v &= 2(\sin 5^{\circ}) + 4(-\cos 15^{\circ}) \\ &\approx 2(0.0872) + 4(-0.9659) \\ &\approx 0.1744 - 3.8636 \\ &\approx -3.6892 \end{aligned}

Similarly, let's evaluate the y-component of the sum $u + v$ :

\begin{aligned} \text{y-component of } u + v &= 2(\cos 5^{\circ}) + 4(\sin 15^{\circ}) \\ &\approx 2(0.9962) + 4(0.2588) \\ &\approx 1.9924 + 1.0352 \\ &\approx 3.0276 \end{aligned}

Conclusion

In this article, we have demonstrated the process of adding two vectors, $u$ and $v$ . We have used the given vectors to calculate the sum $u + v$ and simplified the components using trigonometric identities. Finally, we have evaluated the components using a calculator or a trigonometric table.

The sum $u + v$ is a vector with x-component approximately equal to -3.6892 and y-component approximately equal to 3.0276.

Final Answer

The final answer is:

\begin{aligned} u + v &\approx \left\langle -3.6892, 3.0276 \right\rangle \end{aligned}$$<br/> **Vector Addition: A Comprehensive Guide** =============================================

Q&A: Vector Addition

Q: What is vector addition?

A: Vector addition is the process of combining two or more vectors to form a new vector. This is done by adding the corresponding components of the vectors.

Q: How do I add two vectors?

A: To add two vectors, you need to add their corresponding components. This means that you add the x-components (the first components) and the y-components (the second components) separately.

Q: What are the steps to add two vectors?

A: The steps to add two vectors are:

Identify the components of the two vectors.
Add the corresponding components of the two vectors.
Simplify the resulting components using trigonometric identities.
Evaluate the simplified components using a calculator or a trigonometric table.

Q: What is the difference between vector addition and scalar multiplication?

A: Vector addition and scalar multiplication are two different operations in vector mathematics. Vector addition involves combining two or more vectors to form a new vector, while scalar multiplication involves multiplying a vector by a scalar (a number) to form a new vector.

Q: Can I add more than two vectors?

A: Yes, you can add more than two vectors. The process is the same as adding two vectors, but you need to add the corresponding components of all the vectors.

Q: What are some common applications of vector addition?

A: Vector addition has many applications in physics, engineering, and computer science. Some common applications include:

Calculating the resultant force of multiple forces acting on an object.
Determining the position and velocity of an object in motion.
Calculating the electric field of multiple charges.
Determining the magnetic field of multiple magnets.

Q: How do I represent a vector in a diagram?

A: To represent a vector in a diagram, you need to draw an arrow that points in the direction of the vector. The length of the arrow represents the magnitude of the vector, and the direction of the arrow represents the direction of the vector.

Q: What are some common mistakes to avoid when adding vectors?

A: Some common mistakes to avoid when adding vectors include:

Failing to add the corresponding components of the vectors.
Not simplifying the resulting components using trigonometric identities.
Not evaluating the simplified components using a calculator or a trigonometric table.
Not representing the vector in a diagram correctly.

Conclusion

In this article, we have answered some common questions about vector addition. We have covered the basics of vector addition, including how to add two vectors, the steps to add two vectors, and some common applications of vector addition. We have also discussed some common mistakes to avoid when adding vectors.

Final Answer

The final answer is:

Vector addition is the process of combining two or more vectors to form a new vector.
To add two vectors, you need to add their corresponding components.
The steps to add two vectors are: identify the components of the two vectors, add the corresponding components of the two vectors, simplify the resulting components using trigonometric identities, and evaluate the simplified components using a calculator or a trigonometric table.
Vector addition has many applications in physics, engineering, and computer science.
To represent a vector in a diagram, you need to draw an arrow that points in the direction of the vector.
Some common mistakes to avoid when adding vectors include failing to add the corresponding components of the vectors, not simplifying the resulting components using trigonometric identities, not evaluating the simplified components using a calculator or a trigonometric table, and not representing the vector in a diagram correctly.