3.2 - Vectors
Vector: A quantity specified
by both magnitude and
direction.
Vectors have both magnitude (how much) and
direction. For example, vectors can be used to supply
traveling instructions. If a pilot is told “Fly 20
kilometers due south,” she is being given a
displacement vector to follow. Its magnitude is 20
kilometers and its direction is south. Vector
magnitudes are positive or zero; it would be confusing
to tell somebody to drive negative 20 kilometers south.
Many of the fundamental quantities in physics are vectors. For instance, displacement,
velocity and acceleration are all vector quantities. Physicists depict vectors with arrows.
The length of the arrow is proportional to the vector’s magnitude, and the arrow points
in the direction of the vector. The horizontal vector in Concept 1 on the right represents
the displacement of a car driving from Acme to Dunsville.
You see two displacement vectors in Concept 1. The displacement vector of a drive
from Acme to Dunsville is twice as long as the displacement vector from Chester to
Dunsville. This is because the distance from Acme to Dunsville is twice that of Chester
to Dunsville.
Even if they do not begin at the same point, two vectors are equal if they have the same
magnitude and direction. For instance, the vector from Chester to Dunsville in Concept
1 represents a displacement of 100 km southeast. That vector could be moved without
changing its meaning. Perhaps it is 100 km southeast from Edwards to Frankville, as
well. A vector’s meaning is defined by its length and direction, not by its starting point.
Now that we have introduced the concept of vectors formally, we will express vector
quantities in boldface. For instance, F represents force, v stands for velocity, and so
on. You will often see F and v, as well, representing the magnitudes of the vectors,
without boldface. Why? Because it is frequently useful to discuss the magnitude of the
force or the velocity without concerning ourselves with its direction. For instance, there
may be several equations that determine the magnitude of a vector quantity like force,
but not its direction.
A spelunker (cave explorer) uses both distance and direction to navigate.Vectors
Magnitude and direction
Represented by arrows
Length proportional to magnitude