Solutions to Linear Algebra & Geometry Problems

 1) Solution: 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.

2) We may write the applicable determinant for the parallelopiped as a  3 x 3 matrix:

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

And we saw that the solution was:

Det M = u1[v2 w3 - v3 w2] - u2 [v1 w3 - v3 w1]

+ u3 [v1 w2 - v2 w1]

Now we will apply this to the case of finding the volume of the parallelopiped shown which has 3 sides spanned by the vectors, u, v and w:

where:

u = (1, 1, 3)

v = (1, 2, -1)

w = (1, 4, 1)

We let the volume be Vol(u, v, w) and:

Vol (u, v, w) = Det [u, v, w]

So that:


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


(1.....1......3)
(1....2.....-1)
(1....4........1)

And Det [u, v, w] =

    =

[2 -(-4) - (1 - (-1)) + 3(4 - 2)] = 6 - (2) + 3(2)

Vol (u, v. w) = 12 - 2 = 10 cubic units