A Vector Is ( X Y ) \binom{x}{y} ( Y X ​ ) . Its Magnitude Is 5, And Its Angle With The X X X -axis Is \tan^{-1}\left(\frac{3}{4}\right ]. Find The Values Of X X X And Y Y Y .

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Introduction

In the Cartesian plane, a vector can be represented as $\binom{x}{y}$, where $x$ and $y$ are the horizontal and vertical components of the vector, respectively. The magnitude of a vector is a measure of its length or size, and it can be calculated using the formula $\sqrt{x^2 + y^2}$. The angle of a vector with the $x$-axis is a measure of its direction, and it can be calculated using the formula $\tan^{-1}\left(\frac{y}{x}\right)$. In this article, we will discuss how to find the values of $x$ and $y$ given the magnitude and angle of a vector.

The Magnitude of a Vector

The magnitude of a vector is a measure of its length or size. It can be calculated using the formula $\sqrt{x^2 + y^2}$. In this problem, the magnitude of the vector is given as 5, so we can write an equation:

$$\sqrt{x^2 + y^2} = 5$$

The Angle of a Vector

The angle of a vector with the $x$-axis is a measure of its direction. It can be calculated using the formula $\tan^{-1}\left(\frac{y}{x}\right)$. In this problem, the angle of the vector is given as $\tan^{-1}\left(\frac{3}{4}\right)$, so we can write an equation:

$$\tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{3}{4}\right)$$

Solving the Equations

To solve the equations, we can start by squaring both sides of the first equation:

$$x^2 + y^2 = 25$$

We can then substitute this expression into the second equation:

$$\tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{3}{4}\right)$$

$$\tan^{-1}\left(\frac{y}{\sqrt{25-y^2}}\right) = \tan^{-1}\left(\frac{3}{4}\right)$$

Using Trigonometric Identities

We can use the trigonometric identity $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ to rewrite the equation:

$$\tan^{-1}\left(\frac{y}{\sqrt{25-y^2}}\right) = \tan^{-1}\left(\frac{3}{4}\right)$$

$$\frac{y}{\sqrt{25-y^2}} = \frac{3}{4}$$

Cross-Multiplying

We can cross-multiply the equation to get:

$$4y = 3\sqrt{25-y^2}$$

Squaring Both Sides

We can square both sides of the equation to get:

$$16y^2 = 9(25-y^2)$$

Expanding the Equation

We can expand the equation to get:

$$16y^2 = 225 - 9y^2$$

Combining Like Terms

We can combine like terms to get:

$$25y^2 = 225$$

Dividing Both Sides

We can divide both sides of the equation by 25 to get:

$$y^2 = 9$$

Taking the Square Root

We can take the square root of both sides of the equation to get:

$$y = \pm 3$$

Substituting the Value of $y$

We can substitute the value of $y$ into the equation $x^2 + y^2 = 25$ to get:

$$x^2 + 9 = 25$$

Solving for $x$

We can solve for $x$ to get:

$$x^2 = 16$$

Taking the Square Root

We can take the square root of both sides of the equation to get:

$$x = \pm 4$$

The Values of $x$ and $y$

We have found the values of $x$ and $y$ to be $x = \pm 4$ and $y = \pm 3$.

Conclusion

In this article, we have discussed how to find the values of $x$ and $y$ given the magnitude and angle of a vector. We have used the formulas for the magnitude and angle of a vector, and we have solved the resulting equations using algebraic and trigonometric techniques. We have found the values of $x$ and $y$ to be $x = \pm 4$ and $y = \pm 3$.

References

• [1] "Vector Calculus" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Further Reading

• [1] "Vector Analysis" by Willard Van Orman Quine
• [2] "Calculus: Early Transcendentals" by James Stewart
• [3] "Linear Algebra and Its Applications" by Gilbert Strang
  

Introduction

In our previous article, we discussed how to find the values of $x$ and $y$ given the magnitude and angle of a vector. In this article, we will answer some frequently asked questions about vectors in the Cartesian plane.

Q: What is a vector in the Cartesian plane?

A: A vector in the Cartesian plane is a mathematical object that has both magnitude (length) and direction. It can be represented as $\binom{x}{y}$, where $x$ and $y$ are the horizontal and vertical components of the vector, respectively.

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

A: The magnitude of a vector can be calculated using the formula $\sqrt{x^2 + y^2}$.

Q: How do I calculate the angle of a vector?

A: The angle of a vector can be calculated using the formula $\tan^{-1}\left(\frac{y}{x}\right)$.

Q: What is the relationship between the magnitude and angle of a vector?

A: The magnitude and angle of a vector are related by the formula $\tan(\theta) = \frac{y}{x}$, where $\theta$ is the angle of the vector.

Q: How do I find the values of $x$ and $y$ given the magnitude and angle of a vector?

A: To find the values of $x$ and $y$ given the magnitude and angle of a vector, you can use the formulas for the magnitude and angle of a vector, and then solve the resulting equations using algebraic and trigonometric techniques.

Q: What are some common mistakes to avoid when working with vectors in the Cartesian plane?

A: Some common mistakes to avoid when working with vectors in the Cartesian plane include:

• Not using the correct formulas for the magnitude and angle of a vector
• Not solving the resulting equations correctly
• Not checking the units of the variables
• Not using the correct trigonometric identities

Q: What are some real-world applications of vectors in the Cartesian plane?

A: Vectors in the Cartesian plane have many real-world applications, including:

• Physics: Vectors are used to describe the motion of objects in space.
• Engineering: Vectors are used to describe the forces and motions of objects in engineering applications.
• Computer Graphics: Vectors are used to describe the positions and orientations of objects in computer graphics.
• Navigation: Vectors are used to describe the positions and directions of objects in navigation systems.

Q: How do I choose the correct trigonometric identity to use in a problem?

A: To choose the correct trigonometric identity to use in a problem, you should consider the following:

• What is the relationship between the variables in the problem?
• What is the goal of the problem?
• What trigonometric identities are relevant to the problem?

Q: What are some common trigonometric identities that I should know?

A: Some common trigonometric identities that you should know include:

• $\sin^2(\theta) + \cos^2(\theta) = 1$
• $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$
• $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$

Q: How do I check my work when working with vectors in the Cartesian plane?

A: To check your work when working with vectors in the Cartesian plane, you should:

• Check the units of the variables
• Check the signs of the variables
• Check the trigonometric identities used
• Check the final answer

Conclusion

In this article, we have answered some frequently asked questions about vectors in the Cartesian plane. We have discussed the formulas for the magnitude and angle of a vector, and we have provided some tips and tricks for working with vectors in the Cartesian plane. We have also discussed some common mistakes to avoid and some real-world applications of vectors in the Cartesian plane.

References

• [1] "Vector Calculus" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Linear Algebra" by Jim Hefferon

Further Reading

• [1] "Vector Analysis" by Willard Van Orman Quine
• [2] "Calculus: Early Transcendentals" by James Stewart
• [3] "Linear Algebra and Its Applications" by Gilbert Strang