Saturday, January 7, 2012

Linear Algebra Applied to Geometry

















One of the more interesting applications of linear algebra is to plane and solid geometry. Most of these applications entail computing the determinant D of a matrix. To recap, given a 2 x 2 matrix say:

(a11.......a12)
(a21......a22)

The determinant D is computed:

D = (a11 x a22) - [(a12) x (a21)]

In geometric applications, one will take the absolute value [x] of the result.

Let's take as an example finding the area of a parallelogram such as shown in Graph 1. This figure is spanned by the vectors (2, 1) and (-4, 5) as shown. The area will then be the determinant D of the matrix formed. This matrix will be such that:

a11 = 2, a12 = 1, a21 = -4 and a22 = 5

Then: D = (a11 x a22) - [(a12) x (a21)] = (2 x 5) - [(1 x (-4)] = 10 + 4 = 14 sq. units

Example (2):

Find the area of the parallelogram as depicted in Graph 2.

In this case, the spanning vectors are (3,2) and (-2, -3), so we have for the elements of the matrix:

a11 = 3, a12 = 2, a21 = -2 and a22 = -3.

Then: D = (a11 x a22) - [(a12) x (a21)] = [(3 x (-3)] - [2 x (-2)] = -9 + 4 = [-5] = 5 sq. units

Note that the abs. value must be taken because the determinant is negative in Ex. (2).


Example (3):

Find the area of a parallelogram such that 3 of its corners are given by the points:

(1,1), (2, -1) and (4, 6)

We obtain the spanning vectors here by taking differences. So, let: A = (1, 1), B = (2, -1) and C = (4, 6). Then:

B - A = [(2 - 1), (-1 -1)] = (1, -2)

C - B = [(4 - 2), (6 - (-1)] = (2, 7)

Then: the matrix is: M =

(1....-2)
(2.....7)

So Det (M) = (1 x 7) - [(-2) x 2]= 7 - (-4) = 7 + 4 = 11 sq. units.

Additional Problems:

1) Find the area of a parallelogram for which three corners have the coordinates: (2, 5), (-1, 4) and (1, 2).

2) The same as (1) except the three corners have coordinates: (1,1), (1, 0) and (2, 3)

3) How would you find the determinant of a 3 x 3 matrix, say:

(u1....u2......u3)
(v1....v2.....v3)
(w1....w2....w3)

Give the value of D in terms of the given elements.

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