i
i
i
i
i
i
i
i
92 5. Linear Algebra
parallelepiped (3D parallelogram; a sheared 3D box) formed by the three vectors
(Figure 5.2). To compute a 2D determinant, we first need to establish a few of its
properties. We note that scaling one side of a parallelogram scales its area by the
same fraction (Figure 5.3):
|(ka)b| = |a(kb)| = k|ab|.
Also, we note that “shearing” a parallelogram does not change its area (Fig-
ure 5.4):
Figure 5.3. Scaling a par-
allelogram along one direc-
tion changes the area in the
same proportion.
|(a + kb)b| = |a(b + ka)| = |ab|.
Finally, we see that the determinant has the following property:
|a(b + c)| = |ab|+ |ac|, (5.1)
because as shown in Figure 5.5 we can “slide” the edge between the two parallel-
ograms over to form a single parallelogram without changing the area of either of
the two original parallelograms.
Now let’s assume a Cartesian representation for a and b:
Figure 5.4. Shearing
a parallelogram does not
change its area. These
four parallelograms have
the same length base and
thus the same area.
|ab| = |(x
a
x + y
a
y)(x
b
x + y
b
y)|
= x
a
x
b
|xx| + x
a
y
b
|xy| + y
a
x
b
|yx|+ y
a
y
b
|yy|
= x
a
x
b
(0) + x
a
y
b
(+1) + y
a
x
b
(−1) + y
a
y
b
(0)
= x
a
y
b
− y
a
x
b
.
This simplification uses the fact that |vv | =0for any vector v, because the
parallelograms would all be collinear with v and thus without area.
In three dimensions, the determinantof three 3D vectorsa, b,andc is denoted
|abc|. With Cartesian representations for the vectors, there are analogous rules
for parallelepipeds as there are for parallelograms, and we can do an analogous
expansion as we did for 2D:
|abc| = |(x
a
x + y
a
y + z
a
z)(x
b
x + y
b
y + z
b
z)(x
c
x + y
c
y + z
c
z)|
= x
a
y
b
z
c
− x
a
z
b
y
c
− y
a
x
b
z
c
+ y
a
z
b
x
c
+ z
a
x
b
y
c
− z
a
y
b
x
c
.
Figure 5.5. The geometry
behind Equation 5.1. Both
of the parallelograms on the
left can be sheared to cover
the single parallelogram on
the right.
As you can see, the computation of determinants in this fashion gets uglier as the
dimension increases. We will discuss less error-prone ways to compute determi-
nants in Section 5.3.
Example. Determinants arise naturally when computing the expression for one
vector as a linear combination of two others—for example, if we wish to express
a vector c as a combination of vectors a and b:
c = a
c
a + b
c
b.