Calculating the Ordered Pair

Oct 13, 2025 by talynt.org 107 views

Introduction

In mathematics, vectors are used to represent quantities with both magnitude and direction. Given two points in a coordinate plane, we can find the vector that represents the direction from one point to the other. In this article, we will learn how to write the ordered pair that represents the vector $\overrightarrow{YZ}$ and find its magnitude.

Understanding Vectors

A vector is a quantity with both magnitude and direction. It can be represented graphically as an arrow in a coordinate plane. The magnitude of a vector is its length, while the direction is the angle it makes with the positive x-axis. Vectors can be added, subtracted, and multiplied, making them a fundamental concept in mathematics and physics.

Writing the Ordered Pair that Represents $\overrightarrow{YZ}$

To write the ordered pair that represents the vector $\overrightarrow{YZ}$ , we need to find the difference between the coordinates of points Y and Z. The coordinates of point Y are (-4, 12), and the coordinates of point Z are (1, 19). The ordered pair that represents the vector $\overrightarrow{YZ}$ is given by:

$\overrightarrow{YZ} = (x_2 - x_1, y_2 - y_1)$

where $(x_1, y_1)$ are the coordinates of point Y and $(x_2, y_2)$ are the coordinates of point Z.

Calculating the Ordered Pair

Substituting the coordinates of points Y and Z into the formula, we get:

$\overrightarrow{YZ} = (1 - (-4), 19 - 12)$

$\overrightarrow{YZ} = (5, 7)$

Therefore, the ordered pair that represents the vector $\overrightarrow{YZ}$ is (5, 7).

Finding the Magnitude of $\overrightarrow{YZ}$

The magnitude of a vector is its length, which can be found using the distance formula. The distance formula is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

where $(x_1, y_1)$ are the coordinates of point Y and $(x_2, y_2)$ are the coordinates of point Z.

Calculating the Magnitude

Substituting the coordinates of points Y and Z into the formula, we get:

$d = \sqrt{(5 - (-4))^2 + (7 - 12)^2}$

$d = \sqrt{(9)^2 + (-5)^2}$

$d = \sqrt{81 + 25}$

$d = \sqrt{106}$

Therefore, the magnitude of the vector $\overrightarrow{YZ}$ is $\sqrt{106}$ .

Conclusion

In this article, we learned how to write the ordered pair that represents the vector $\overrightarrow{YZ}$ and find its magnitude. We used the difference between the coordinates of points Y and Z to find the ordered pair and the distance formula to find the magnitude. The ordered pair that represents the vector $\overrightarrow{YZ}$ is (5, 7), and its magnitude is $\sqrt{106}$ .

Applications of Vectors

Vectors have numerous applications in mathematics and physics. Some of the applications include:

Physics: Vectors are used to represent forces, velocities, and accelerations in physics.
Engineering: Vectors are used to represent the motion of objects in engineering.
Computer Graphics: Vectors are used to represent the position and orientation of objects in computer graphics.
Navigation: Vectors are used to represent the direction and distance between two points in navigation.

Real-World Examples

Vectors have numerous real-world examples. Some of the examples include:

GPS Navigation: Vectors are used to represent the direction and distance between two points in GPS navigation.
Flight Planning: Vectors are used to represent the direction and distance between two points in flight planning.
Surveying: Vectors are used to represent the direction and distance between two points in surveying.
Computer-Aided Design (CAD): Vectors are used to represent the position and orientation of objects in CAD.

Final Thoughts

In conclusion, vectors are a fundamental concept in mathematics and physics. They have numerous applications in various fields, including physics, engineering, computer graphics, and navigation. The ordered pair that represents the vector $\overrightarrow{YZ}$ is (5, 7), and its magnitude is $\sqrt{106}$ . We hope this article has provided a comprehensive understanding of vectors and their applications.

Introduction

Vectors are a fundamental concept in mathematics and physics, and they have numerous applications in various fields. In this article, we will answer some of the most frequently asked questions about vectors.

Q: What is a vector?

A: A vector is a quantity with both magnitude and direction. It can be represented graphically as an arrow in a coordinate plane.

Q: How do I represent a vector?

A: A vector can be represented as an ordered pair of numbers, (x, y), where x is the horizontal component and y is the vertical component.

Q: What is the difference between a vector and a scalar?

A: A scalar is a quantity with only magnitude, while a vector has both magnitude and direction.

Q: How do I add two vectors?

A: To add two vectors, you need to add their corresponding components. For example, if you have two vectors, (x1, y1) and (x2, y2), the sum of the two vectors is (x1 + x2, y1 + y2).

Q: How do I subtract two vectors?

A: To subtract two vectors, you need to subtract their corresponding components. For example, if you have two vectors, (x1, y1) and (x2, y2), the difference of the two vectors is (x1 - x2, y1 - y2).

Q: What is the magnitude of a vector?

A: The magnitude of a vector is its length, which can be found using the distance formula.

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

A: To find the magnitude of a vector, you need to use the distance formula, which is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Q: What is the direction of a vector?

A: The direction of a vector is the angle it makes with the positive x-axis.

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

A: To find the direction of a vector, you need to use the inverse tangent function, which is given by:

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

Q: What is the dot product of two vectors?

A: The dot product of two vectors is a scalar quantity that represents the amount of "similarity" between the two vectors.

Q: How do I find the dot product of two vectors?

A: To find the dot product of two vectors, you need to multiply their corresponding components and add them together. For example, if you have two vectors, (x1, y1) and (x2, y2), the dot product of the two vectors is:

$(x1 \cdot x2) + (y1 \cdot y2)$

Q: What is the cross product of two vectors?

A: The cross product of two vectors is a vector quantity that represents the amount of "perpendicularity" between the two vectors.

Q: How do I find the cross product of two vectors?

A: To find the cross product of two vectors, you need to use the following formula:

$\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \end{vmatrix}$

where $\mathbf{i}$ , $\mathbf{j}$ , and $\mathbf{k}$ are the unit vectors in the x, y, and z directions, respectively.

Q: What is the unit vector of a vector?

A: The unit vector of a vector is a vector with a magnitude of 1 that points in the same direction as the original vector.

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

A: To find the unit vector of a vector, you need to divide the vector by its magnitude. For example, if you have a vector, (x, y), the unit vector of the vector is:

$\frac{(x, y)}{\sqrt{x^2 + y^2}}$

Conclusion

In this article, we have answered some of the most frequently asked questions about vectors. We hope this article has provided a comprehensive understanding of vectors and their applications.