Converting Vectors to Rectangular Form

Oct 13, 2025 by pinkandtrash.com 17 views

**The Vector Addition Problem: A Comprehensive Guide** =====================================================

Introduction

In this article, we will delve into the world of vector addition and explore the process of finding the magnitude and direction angle of the resultant vector. We will use the given information about two vectors, ${$ \mathbf{v} $}$ and ${$ \mathbf{w} $}$, to calculate the magnitude and direction angle of their sum.

Understanding Vectors

A vector is a quantity with both magnitude and direction. It can be represented graphically as an arrow in a coordinate system, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction of the vector.

The Given Vectors

We are given two vectors:

${$ \mathbf{v} $}$ has magnitude 9 and direction angle ${$ 298^\circ $}$.
${$ \mathbf{w} $}$ has magnitude 6 and direction angle ${$ 90^\circ $}$.

Vector Addition

To add two vectors, we can use the following formula:

${$ \mathbf{a} + \mathbf{b} = \sqrt{a^2 + b^2 + 2ab\cos\theta} $}$

where ${$ \mathbf{a} $}$ and ${$ \mathbf{b} $}$ are the two vectors being added, and ${$ \theta $}$ is the angle between them.

However, since we are given the direction angles of the vectors, we can use a different approach. We can convert the vectors to their rectangular form using the following formulas:

${$ x = r\cos\theta $}$

${$ y = r\sin\theta $}$

where ${$ r $}$ is the magnitude of the vector and ${$ \theta $}$ is the direction angle.

Converting Vectors to Rectangular Form

Let's convert the given vectors to their rectangular form:

${$ \mathbf{v} = 9\cos(298^\circ) + 9\sin(298^\circ) $}$

${$ \mathbf{w} = 6\cos(90^\circ) + 6\sin(90^\circ) $}$

Using a calculator, we can find the values of the rectangular components:

${$ \mathbf{v} = -7.66 + 0.66i $}$

${$ \mathbf{w} = 0 + 6i $}$

Adding the Vectors

Now that we have the rectangular form of the vectors, we can add them:

${$ \mathbf{v} + \mathbf{w} = (-7.66 + 0.66i) + (0 + 6i) $}$

${$ \mathbf{v} + \mathbf{w} = -7.66 + 6.66i $}$

Finding the Magnitude and Direction Angle

To find the magnitude of the resultant vector, we can use the following formula:

${$ r = \sqrt{x^2 + y^2} $}$

where ${$ x $}$ and ${$ y $}$ are the rectangular components of the vector.

Plugging in the values, we get:

${$ r = \sqrt{(-7.66)^2 + (6.66)^2} $}$

${$ r = \sqrt{58.35 + 44.35} $}$

${$ r = \sqrt{102.7} $}$

${$ r = 10.13 $}$

To find the direction angle, we can use the following formula:

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

where ${$ x $}$ and ${$ y $}$ are the rectangular components of the vector.

Plugging in the values, we get:

${$ \theta = \tan^{-1}\left(\frac{6.66}{-7.66}\right) $}$

${$ \theta = \tan^{-1}(-0.87) $}$

${$ \theta = -40.9^\circ $}$

Conclusion

In this article, we have explored the process of finding the magnitude and direction angle of the resultant vector when two vectors are added. We have used the given information about the vectors ${$ \mathbf{v} $}$ and ${$ \mathbf{w} $}$ to calculate the magnitude and direction angle of their sum.

Frequently Asked Questions

Q: What is the magnitude of the resultant vector?

A: The magnitude of the resultant vector is 10.13.

Q: What is the direction angle of the resultant vector?

A: The direction angle of the resultant vector is -40.9°.

Q: How do I convert a vector to its rectangular form?

A: To convert a vector to its rectangular form, you can use the following formulas:

${$ x = r\cos\theta $}$

${$ y = r\sin\theta $}$

where ${$ r $}$ is the magnitude of the vector and ${$ \theta $}$ is the direction angle.

Q: How do I add two vectors?

A: To add two vectors, you can use the following formula:

${$ \mathbf{a} + \mathbf{b} = \sqrt{a^2 + b^2 + 2ab\cos\theta} $}$

where ${$ \mathbf{a} $}$ and ${$ \mathbf{b} $}$ are the two vectors being added, and ${$ \theta $}$ is the angle between them.

Q: What is the formula for finding the magnitude of a vector?

A: The formula for finding the magnitude of a vector is:

${$ r = \sqrt{x^2 + y^2} $}$

where ${$ x $}$ and ${$ y $}$ are the rectangular components of the vector.

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

A: The formula for finding the direction angle of a vector is:

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

where ${$ x $}$ and ${$ y $}$ are the rectangular components of the vector.