# Finding area of the affine translation of a rectangle

• 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!

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.

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.

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!