Introduction to Vectors 2

One of the most important concepts in math is the vector. Vaguely, we define a vector by a
direction and a magnitude. Graphically, we represent a vector as a directed line segment
from an initial point A to a terminal point B. The magnitude is given by the length of this
segment. We say that two vectors are equal if they have the same magnitude and direction. Hence
the same vector can be drawn from many different starting points. We usually denote a vector by
a bold, lowercase letter, v, or a letter with an arrow on top, ~v.
Analytically, we usually express vectors in component form. We do this by considering the
vector with the origin as its initial point and write the coordinates of its terminal point. For
example, in 2D,
v = hv1, v2i
and in 3D,
v = hv1, v2, v3i
Since the vector hv1, v2i has 0 as its inital point and (v1, v2) as its terminal point, its magnitude
is
kvk = qv2
1 + v2
2
Similarly, a 3D vector in component form has magnitude
kvk = qv2
1 + v2
2 + v2
3
The only vector with 0 magnitude is the zero vector, with all components equal to 0.
Example 1. Find the component form and magnitude of the vector with (4, 2,?1) as its initial
point and (?2, 3, 0) as its terminal point.
The component form of the vector is given by
h?2 ? 4, 3 ? 2, 0 ? ?1i = h?6, 1, 1i
Its magnitude is given by
kvk = ?36 + 1 + 1 = ?38
Example 2 (From Textbook). A small cart is being pulled along a smooth horizontal floor with a
20-lb force F making a 45? angle with the floor. What is the horizontal component of the force?
1
We have here a different way to represent a vector, but we still have a magnitude (20 lbs) and
a direction (45?). The appropriate way to attack the problem is to graph the situation with the
cart at the origin and the ground on the x-axis. Draw the force vector with magnitude 20 making
a 45? angle with the x-axis. We now see that the x-component of the force is
20 cos(45?) = 14.14
The force vector is in fact
h14.14, 14.14i
2 Operations on Vectors
When we are talking about vectors, a real number is called a scalar