3 MOTION IN 3 DIMENSIONS 3.11 Projectile motions|s| (^) a. s = |a| |s| cos
a
|a|
|s| cos
Figure 15: The scalar product
vector s, and θ is the angle subtended between these two vectors, then
a·s = |a| |s| cos θ. (3.39)
In other words, the scalar product of vectors a and s equals the product of the
length of vector a times the length of that component of vector s which lies in the
same direction as vector a. It immediately follows that if two vectors are mutually
perpendicular (i.e., θ = 90 ◦) then their scalar product is zero. Furthermore, the
scalar product of a vector with itself is simply the magnitude squared of that
vector [this is immediately apparent from Eq. (3.38)]:
a·a = |a|^2 = a^2. (3.40)
It is also apparent from Eq. (3.38) that a· s = s· a, and a·(b + c) = a·b + a·c, and
a·(λs) = λ(a· s).
Incidentally, Eq. (3.37) is obtained by taking the scalar product of Eq. (3.36)
with itself, taking the scalar product of Eq. (3.35) with a, and then eliminating t.
1.24 Projectile motion
As a simple illustration of the concepts introduced in the previous subsections, let
us examine the following problem. Suppose that a projectile is launched upward
from ground level, with speed v 0 , making an angle θ with the horizontal. Neglect-
ing the effect of air resistance, what is the subsequent trajectory of the projectile?