Greenwood press

VECTORS
155
Vectors showing James 
j and
Sarah
s pulling on a wagon. Their
resultant force is 
r, indicated by the
diagonal of the parallelogram. 
Vector diagram for an airplane headed
northeast that is blown off course by a
wind 10° south of east.

---

law of cosines 
r
2
= 400
2
+ 90
2
− 2(400)(90) cos 125
◦
. The length of 
r is about
458 miles per hour. The angle between 
a and  r is about 9.3°. So the direction
would be about 
45
◦
− 9.3
◦
= 35.3
◦
north of east. Even though the airplane
would be pointed northeast, from the ground it would appear to be traveling only
35.3° north of east. (See Triangle Trigonometry.)
When vectors are written as an ordered pair, the length is written first, and
the angle second. (See Polar Coordinates.) Sarah’s vector would be written as
s = [25, 20
◦
]; James’ vector would be  j = [15, 0
◦
]. The brackets indicate that the
vector is written in polar-coordinate form. The lengths of the vectors are written
with the absolute value sign. The length of Sarah’s vector would be 
| s| = 25.
Polar form is a natural way of presenting force vectors, but the algebra of vec-
tors is easier to work with in Cartesian-coordinate form (x,y). This is called the
component form. To convert a vector in polar form 
v = [d, θ] to component form,
use the formulas 
x = d cos θ and y = d sin θ. Sarah’s polar vector would be
s = (25 cos 20
◦
, 25 sin 20
◦
) ≈ (23.50, 8.55).To reconstruct the length of Sarah’s
vector from component form, use the Pythagorean theorem: 
| s| =
(25 cos 20
◦
)
2
+ (25 sin 20
◦
)
2
= 25.
The addition of vectors in component form is done by the addition of coor-
dinates. If 
v = (a, b) and 
w = (c, d), the parallelogram law requires that the vec-
tor sum be 
v + 
w = (a + c, b + d). Component form makes it easier to handle
problems involving gravity. If a golf ball is hit with an impact of 70 meters/sec-
ond at a 30° angle, the distance of the ball (ignoring wind resistance and gravity)
is given by the vector 
b = [70t, 30
◦
], where time t is given in seconds. The com-
ponent form is 
b = (70t cos 30
◦
, 70t sin 30
◦
). A graph would show the golf ball
traveling upwards into space at an angle of 30° from the ground. However, grav-
ity provides a force vector that reduces vertical distance as 
g = (0,–4.9t
2
). The
vector addition of the ball and gravity gives a parabolic path produced by
b +  g = (70t cos 30
◦
, 70t sin 30
◦
− 4.9t
2
). Algebra can be used to determine
how far the ball has traveled horizontally when it hits the ground. (See Angle for
computations of the path of a projectile.) Vector descriptions of motions and
forces are used to describe the collisions of atomic particles, the interaction of
chemical substances, and the movements of stars and galaxies.
Component form has operations that are somewhat like multiplication, but
yet different. The dot product of two vectors is given by 
v • 
w = (ac, bd), where
v = (a, b) and 
w = (c, d). Lengthening a vector by a scale factor k is given by
k v = (ka, kb). The dot product is used in the formula for the cosine of an angle 
between two vectors: 
cos θ =

v• 
w
|v|| 
w|
. The effectiveness of component-form vec-
tors comes when vectors operate in more dimensions. For three-dimensional
space, the dot product of 
v = (v
1
, v
2
, v
3
) and 
w = (w
1
, w
2
, w
3
) is  v • 
w =
(v
1
w
1
, v
2
w
2
, v
3
w
3
), an easy-to-remember extension of the two-component
model. Further, the equation for the cosine of the angle between two vectors
looks exactly the same, even though there is an additional dimension.