A Vector Is ( X Y ) \binom{x}{y} ( Y X ​ ) . Its Magnitude Is 5, And The 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 mathematics, a vector is a quantity with both magnitude and direction. It is often represented as an arrow in a 2D or 3D space. In this article, we will focus on a vector in 2D space, represented as $\binom{x}{y}$. The magnitude of the vector is given as 5, and the angle with the $x$-axis is $\tan^{-1}\left(\frac{3}{4}\right)$. Our goal is to find the values of $x$ and $y$.

Understanding the Magnitude and Angle of a Vector

The magnitude of a vector is a measure of its length or size. It is denoted by the symbol $|\vec{v}|$ or $||\vec{v}||$. In this case, the magnitude of the vector is given as 5. The angle of a vector is the angle it makes with the $x$-axis. It is denoted by the symbol $\theta$.

Using Trigonometry to Find the Values of $x$ and $y$

We can use trigonometry to find the values of $x$ and $y$. The magnitude of the vector is given by the formula:

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

We are given that the magnitude of the vector is 5, so we can set up the equation:

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

Squaring both sides of the equation, we get:

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

We are also given that the angle of the vector is $\tan^{-1}\left(\frac{3}{4}\right)$. We can use the formula:

$$\tan\theta = \frac{y}{x}$$

Substituting the value of $\theta$, we get:

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

Simplifying the equation, we get:

$$\frac{3}{4} = \frac{y}{x}$$

Cross-multiplying, we get:

$$3x = 4y$$

Solving the System of Equations

We now have two equations:

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

$$3x = 4y$$

We can solve this system of equations using substitution or elimination. Let's use substitution. Rearranging the second equation, we get:

$$y = \frac{3}{4}x$$

Substituting this expression for $y$ into the first equation, we get:

$$25 = x^2 + \left(\frac{3}{4}x\right)^2$$

Simplifying the equation, we get:

$$25 = x^2 + \frac{9}{16}x^2$$

Combine like terms:

$$25 = \frac{25}{16}x^2$$

Multiply both sides by 16:

$$400 = 25x^2$$

Divide both sides by 25:

$$16 = x^2$$

Take the square root of both sides:

$$x = \pm 4$$

Finding the Value of $y$

Now that we have found the value of $x$, we can find the value of $y$. Substituting the value of $x$ into the equation:

$$y = \frac{3}{4}x$$

We get:

$$y = \frac{3}{4}(4)$$

Simplifying the equation, we get:

$$y = 3$$

Conclusion

In this article, we have found the values of $x$ and $y$ for a vector in 2D space. The magnitude of the vector is 5, and the angle with the $x$-axis is $\tan^{-1}\left(\frac{3}{4}\right)$. We used trigonometry to find the values of $x$ and $y$. The final values are $x = \pm 4$ and $y = 3$.

Final Answer

The final answer is $\boxed{\binom{4}{3}}$ and $\boxed{\binom{-4}{3}}$.

Introduction

In our previous article, we discussed how to find the values of $x$ and $y$ for a vector in 2D space. The magnitude of the vector is 5, and the angle with the $x$-axis is $\tan^{-1}\left(\frac{3}{4}\right)$. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the magnitude of a vector?

A: The magnitude of a vector is a measure of its length or size. It is denoted by the symbol $|\vec{v}|$ or $||\vec{v}||$. In this case, the magnitude of the vector is given as 5.

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

A: To find the magnitude of a vector, you can use the formula:

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

Q: What is the angle of a vector?

A: The angle of a vector is the angle it makes with the $x$-axis. It is denoted by the symbol $\theta$.

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

A: To find the angle of a vector, you can use the formula:

$$\tan\theta = \frac{y}{x}$$

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:

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

$$\tan\theta = \frac{y}{x}$$

Q: How do I find the values of $x$ and $y$ for a vector in 2D space?

A: To find the values of $x$ and $y$ for a vector in 2D space, you can use the following steps:

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

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

2. Find the angle of the vector using the formula:

$$\tan\theta = \frac{y}{x}$$

3. Use the magnitude and angle to find the values of $x$ and $y$.

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

A: Some common mistakes to avoid when working with vectors include:

• Not using the correct formula for the magnitude and angle of a vector
• Not using the correct units for the magnitude and angle of a vector
• Not checking the signs of the values of $x$ and $y$

Q: How do I check the signs of the values of $x$ and $y$?

A: To check the signs of the values of $x$ and $y$, you can use the following steps:

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

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

2. Find the angle of the vector using the formula:

$$\tan\theta = \frac{y}{x}$$

3. Use the magnitude and angle to find the signs of the values of $x$ and $y$.

Q: What are some real-world applications of vectors?

A: Some real-world applications of vectors include:

• 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 Science: Vectors are used to describe the positions and velocities of objects in computer graphics and game development.

Conclusion

In this article, we have answered some frequently asked questions related to finding the values of $x$ and $y$ for a vector in 2D space. We have also discussed some common mistakes to avoid when working with vectors and some real-world applications of vectors.