Greenwood press

VECTORS
157

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indicates that 70 percent of the variability in math scores is accounted for by the
variability in science scores. When statisticians work with many scores, they
examine the correlations among all the variables to determine how the number
of dimensions of the original space can be reduced to fewer, stronger, and more
interpretable dimensions.
In the case of three dimensions, the operation of cross product provides a
way to compute perpendiculars to planes. The cross product of 
v = (v
1
, v
2
, v
3
)
and
w = (w
1
, w
2
, w
3
) is defined as  v
× 
w = (v
2
w
3
− v
3
w
2
, v
3
w
1
− v
1
w
3
,
v
1
w
2
− v
2
w
1
). The cross product is a vector. Its relationship to the plane formed
by
v and 
w is shown in the figure. The cross product is said to be orthogonal to
the plane.
The cross product is computed for surfaces of airplanes or boat hulls. The
direction of air or water currents across the surfaces is modeled by the angles that
the currents make with vectors that are orthogonal to the surface. This is not a
recent concept. Sketches in the notebooks of the Wright brothers one hundred
years ago show computation of vector forces on the different wings they tried
before achieving the first airplane flight. A spinning wheel, like the disk in a
gyroscope, produces a force called torque. This is a force that is perpendicular to
the plane of rotation. If you ride a bike very fast, you will feel resistance as you
try to tilt the bicycle to the left or right. The torque produced by the spinning
wheels will try to maintain its direction, so you must use some pressure to pro-
duce a tilt. If you are traveling slowly, the torque isn’t very strong, so it is easy
to tilt the bike and fall. Large cruise ships have gyroscopes with heavy wheels
that spin rapidly. The torque produces a force that counters the movement from
waves, making for a smoother ride for passengers.
Computer-graphic programmers use orthogonal vectors to determine how
light sources would hit surfaces visible in a computer game or architectural image.
The angles between the light rays from an external source to orthogonal vectors
on the surface are computed. If the angles are close to 0°, then the light will be
shown at full intensity. If close to 90°, then the light is reaching the surface with
minimal intensity. The vector computations (the vector-graphic phase) are then
transferred into the display device as light intensity and color for the different
points (pixels) that would be visible. This is the raster graphic phase. By control-
ling the brilliance of pixels on the screen according to vector computations, com-
puter-graphic designers present realistic scenes to the viewer. Some computer
files store images as vectors (the rules that create the image), and some files keep
the bitmap of the image (a snapshot of the pixel intensity). Postscript files contain