Finding area of the affine translation of a rectangle

• yomakaflo
In summary, the area of the affin translation of the rectangle R is 76, which is obtained by multiplying the original area of R (4) by the determinant of the translation matrix (19). The vectors along the sides of the square should be used for the cross product, not the intervals along the x and y axes.
yomakaflo

Homework Statement

Given a rectangle R=[1,3] x [2,4], and the affin translation F : R^2 -> R^2 defined by F(x,y) = (1,3) + A*(x,y), where A is the 2x2 matrix (2 , 7 ; 3 , 1), what is the area of the affin transelation of the rectangle R?

The Attempt at a Solution

When I cross the vectors of R I get the scalar 2. Is this the area of R before we transelate it? The determinant of A equals 19, and 2*19=38. So this is my answer and it is wrong. Right answer is 76, so I guess the area of R before translation should be 76/det(A)=4. Where am I wrong?

The area of R is obiously 2x2=4 and it must be multiplied by 19 to get the translated area. What are the "vectors of R" that you crossed? Show us that.

R = [1,3] cross [2,4], so I think the vectors of R are [1,3] and [2,4] . When they are crossed the product is abs(4-6) = 2. I want the answer to be 4, but I don't know how!

yomakaflo said:
R = [1,3] cross [2,4], so I think the vectors of R are [1,3] and [2,4] . When they are crossed the product is abs(4-6) = 2. I want the answer to be 4, but I don't know how!

Those are not the correct vectors to cross. You want the vectors along the sides of the square.

yomakaflo said:
R = [1,3] cross [2,4], so I think the vectors of R are [1,3] and [2,4] . When they are crossed the product is abs(4-6) = 2. I want the answer to be 4, but I don't know how!
In addition to what LCKurtz said, those aren't even vectors -- they are intervals along the x and y axes. Also, I don't know what you are doing when you say you are "crossing" these vectors. The vector cross product is defined for vectors in R3.

Mark44 said:
In addition to what LCKurtz said, those aren't even vectors -- they are intervals along the x and y axes. Also, I don't know what you are doing when you say you are "crossing" these vectors. The vector cross product is defined for vectors in R3.

Okey, I misunderstood R = [1,3] x [2,4]. Then i makes sense that the area of R is 4 and 4*19=76 after the translation. Thanks!

1. How do you find the area of an affine translation of a rectangle?

The area of an affine translation of a rectangle can be found by first finding the area of the original rectangle, and then applying the same transformation to the length and width of the rectangle. Finally, the area can be calculated by multiplying the transformed length and width together.

2. What is an affine transformation?

An affine transformation is a type of geometric transformation that preserves parallel lines and ratios of distances. It includes operations such as translation, rotation, reflection, and dilation.

3. Can the area of an affine translation of a rectangle be negative?

No, the area of a rectangle cannot be negative. It is always a positive value, representing the amount of space inside the rectangle.

4. How is an affine translation different from a simple translation?

In a simple translation, the shape is moved to a new location without any changes to its size or orientation. In an affine translation, the shape is not only moved, but also undergoes a transformation such as rotation, reflection, or dilation.

5. What is the importance of finding the area of an affine translation of a rectangle?

Finding the area of an affine translation of a rectangle is important in many fields, including engineering, architecture, and computer graphics. It allows us to accurately measure and compare the size of shapes that have undergone transformations, which is essential for many practical applications.

• Calculus and Beyond Homework Help
Replies
14
Views
685
• Calculus and Beyond Homework Help
Replies
1
Views
600
• Calculus and Beyond Homework Help
Replies
7
Views
292
• Calculus and Beyond Homework Help
Replies
6
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
11
Views
2K
• Calculus and Beyond Homework Help
Replies
4
Views
2K
• Calculus and Beyond Homework Help
Replies
14
Views
3K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
1K