We now look at linear algebra applied to solid geometry. In the last set of solutions we saw the clue on how to proceed, which is to find the determinant of a 3 x 3 matrix, viz.
(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)
Now, 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....-1)
{4.....1) -
(1....-1)
(1.....1) +
3 (1....2)
(1.....4)
=
[2 -(-4) - (1 - (-1)) + 3(4 - 2)] = 6 - (2) + 3(2)
And: Vol (u, v. w) = 12 - 2 = 10 cubic units
Problems:
1) For a similar solid to that shown, but with vectors:
u = (1, -1, 4)
v = (1, 1, 0)
w = (-1, 2, 5)
Find: Vol (u, v, w)
2) Show that for the spanning vectors:
u = (-2, 2, 1)
v = (0, 1, 0)
w = (-4, 3, 2)
Vol (u, v, w) = 0
(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)
Now, 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....-1)
{4.....1) -
(1....-1)
(1.....1) +
3 (1....2)
(1.....4)
=
[2 -(-4) - (1 - (-1)) + 3(4 - 2)] = 6 - (2) + 3(2)
And: Vol (u, v. w) = 12 - 2 = 10 cubic units
Problems:
1) For a similar solid to that shown, but with vectors:
u = (1, -1, 4)
v = (1, 1, 0)
w = (-1, 2, 5)
Find: Vol (u, v, w)
2) Show that for the spanning vectors:
u = (-2, 2, 1)
v = (0, 1, 0)
w = (-4, 3, 2)
Vol (u, v, w) = 0
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