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?

Homework Equations

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?

 

[LCKurtz]

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.

 

[yomakaflo]

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!

 

[LCKurtz]

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.

 

[Mark44]

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.

 

[yomakaflo]

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!