Determining the Translation Vector

Oct 13, 2025 by windailys.com 15 views

Introduction

In mathematics, a translation is a fundamental concept that involves moving a point or a shape from one location to another without changing its size or orientation. Given a set of points and their corresponding translated positions, we can identify a general rule that describes the translation mapping. In this article, we will analyze the given points and determine the translation mapping that applies to them.

The Given Points

We are given three points and their corresponding translated positions:

X(-3,-7) → X'(3,-3)
Y(-8,-8) → Y'(-3,-4)
Z(-7,-5) → Z'(-1,-1)

Translation Mapping

A translation mapping is a function that takes a point in the original coordinate system and maps it to a corresponding point in the translated coordinate system. The general rule for a translation mapping is:

If the original point is (x, y), then the translated point is (x + a, y + b)

where (a, b) is the translation vector.

Determining the Translation Vector

To determine the translation vector, we need to find the difference between the original and translated points. Let's calculate the differences for each point:

X(-3,-7) → X'(3,-3)
- Δx = 3 - (-3) = 6
- Δy = -3 - (-7) = 4
Y(-8,-8) → Y'(-3,-4)
- Δx = -3 - (-8) = 5
- Δy = -4 - (-8) = 4
Z(-7,-5) → Z'(-1,-1)
- Δx = -1 - (-7) = 6
- Δy = -1 - (-5) = 4

Analyzing the Translation Vector

From the calculations above, we can see that the differences in the x-coordinates (Δx) are 6, 5, and 6, respectively. Similarly, the differences in the y-coordinates (Δy) are 4, 4, and 4, respectively. This suggests that the translation vector is (6, 4).

Conclusion

Based on the analysis above, we can conclude that the general rule for the translation mapping is:

If the original point is (x, y), then the translated point is (x + 6, y + 4)

This rule applies to all three points given in the problem. Therefore, the translation mapping that describes the given points is a translation by 6 units in the x-direction and 4 units in the y-direction.

Example Use Case

Suppose we have a point P(2, 3) and we want to translate it using the same translation mapping. We can apply the rule by adding 6 to the x-coordinate and 4 to the y-coordinate:

P(2, 3) → P'(2 + 6, 3 + 4) = P'(8, 7)

Therefore, the translated point P' is (8, 7).

Conclusion

In this article, we analyzed the given points and determined the translation mapping that applies to them. We found that the translation vector is (6, 4) and the general rule for the translation mapping is:

If the original point is (x, y), then the translated point is (x + 6, y + 4)

Introduction

In our previous article, we analyzed the given points and determined the translation mapping that applies to them. We found that the translation vector is (6, 4) and the general rule for the translation mapping is:

If the original point is (x, y), then the translated point is (x + 6, y + 4)

In this article, we will provide a Q&A guide to help you understand the translation mapping and its applications.

Q: What is a translation mapping?

A: A translation mapping is a function that takes a point in the original coordinate system and maps it to a corresponding point in the translated coordinate system.

Q: How do I determine the translation vector?

A: To determine the translation vector, you need to find the difference between the original and translated points. You can do this by subtracting the x-coordinates and y-coordinates of the original point from the x-coordinates and y-coordinates of the translated point.

Q: What is the general rule for a translation mapping?

A: The general rule for a translation mapping is:

If the original point is (x, y), then the translated point is (x + a, y + b)

where (a, b) is the translation vector.

Q: How do I apply the translation mapping to a point?

A: To apply the translation mapping to a point, you need to add the translation vector to the coordinates of the point. For example, if the original point is (2, 3) and the translation vector is (6, 4), the translated point would be (2 + 6, 3 + 4) = (8, 7).

Q: Can I use the translation mapping to translate a shape?

A: Yes, you can use the translation mapping to translate a shape. The translation mapping will move the shape from its original position to its new position without changing its size or orientation.

Q: How do I find the inverse of a translation mapping?

A: To find the inverse of a translation mapping, you need to reverse the translation vector. For example, if the translation vector is (6, 4), the inverse translation vector would be (-6, -4).

Q: Can I use the translation mapping to solve real-world problems?

A: Yes, you can use the translation mapping to solve real-world problems. For example, you can use the translation mapping to determine the position of a moving object, to calculate the distance between two points, or to find the area of a translated shape.

Example Problems

Find the translated point of P(2, 3) using the translation mapping (6, 4).
Determine the translation vector for the points A(1, 2) and A'(7, 8).
Find the inverse of the translation mapping (6, 4).
Use the translation mapping to calculate the distance between the points P(2, 3) and Q(8, 7).

Solutions

P(2, 3) → P'(2 + 6, 3 + 4) = P'(8, 7)
Δx = 7 - 1 = 6 Δy = 8 - 2 = 6 The translation vector is (6, 6).
The inverse translation vector is (-6, -4).
The distance between P(2, 3) and Q(8, 7) is √((8 - 2)^2 + (7 - 3)^2) = √(36 + 16) = √52.

Conclusion

In this article, we provided a Q&A guide to help you understand the translation mapping and its applications. We covered topics such as determining the translation vector, applying the translation mapping to a point, and finding the inverse of a translation mapping. We also provided example problems and solutions to help you practice your skills.