University:
University of Arizona
Course:
MATH 413 | Linear Algebra
Academic year:

2023
Views:

77

Pages:

5
Author:

Tressa Z.

Hour 28 Area, Volume, Cross product, and the Determinant No thoughts just webes The determinant of a matrix A can also be thought of as giving the signed area of certain 2d shapes, or the signed volume of certain 3d solids. Let's see this more precisely. 1. Let's start with a 2 x 2 matrix A with columns u and 7, so A = [i]. If we write A = นี= and = The vectors นี้ and ✔ define a parallelogram (see picture): [], then cd are of 1 green : (base) (height) find vectors AB & AC area of 1 blue D: (a+b)(c) 2 groom (a) What is the area of the entire shaded (in blue, green and gray) gegion? (a+b)(c+d) bc rectangle dove ne popla b) Use that to find the area of the parallelogram ( defined by and 7. are of = ara of rectangle-2 (area of green)-2(area (area of blue 1) = ac+ad+bc+bd-R(c+a)(b)/2-8 (a+b)(c)2 actad + bc th bcbd-ac-bc = ad-be = det (A) determiument (c) What if the order of 7 and i are switched? How does that change det: (A), and how does that change the area of the parallelogram? area is still same since is the lame, BUT det([2]) = bc-ad = -(ab-bc) = - det (A)... The area of a parallelogram given by vectas tu, is I det ([]) (absolute value) (d) What if the vectors u and i were parallel? What would be the area then, and what would be the determinant? To find the volume of the parallelepiped, our strategy is first to find the area of the base, and then to multiply that by the height of the parallelepiped. 3. First, let's find the area of the base. We were able to do that for a parallelogram last time, using a determinant, but that was in R. Now our parallelogram is in R³, so we'll need a different technique. What is the area of the parallelogram, if we know the vectors are separated by an angle 6? What quantity from the previous page is this area related to? Dungle between w e height Leight = lwll sin aven = base = height= "wll Kullan (vs payel, area of in or: area below 4. Draw where the cross product (remember its a vector) would go on the pictures. Denote the angle between that vector and as a (the Greek letter "alpha"). Use some more geometry to find the height of the parallelepiped paralele po So wxw I to base (to, w) and & is the angle blow (üxů) anh vector in that lave height Hall cos X So: volume = (area of base) (height of parallelepiped) = luxwll-lul cos a How might we use a determinant to compute a tripin scalar product? A triple scalor product is: u. (ix) ↑ expand along 1st coln V₂ W2 Va ws 1st entry of the vector 2nd enting of tow ind entry of to compute u. (üxw), find (and take absolute value of that to oet volume)

Area, Volume, Cross Product, and the Determinant

Related Documents

Description

Related Documents

Free up your schedule!

Take 5 seconds to unlock