Vectors Outline
Objective: Understand vector, vector components, scalar,
magnitude, scalar multiplication, direction,adding vectors
(algebraically and graphically), subtracting vectors (algebraically
and graphically), unit vec-tors.
Position Vector
The position vector, ~r, from the origin to some point located
at (x, y, z) is written as
~r =< rx, ry, rz > (1)
where rx, ry, rz are the components of the position vector.
Magnitude of a vector
The magnitude of a vector is determined by the Pythagorean
theorem. For example, the magnitude of aposition vector is
|~r| =(r2x + r2y + r2z) (2)
and represents the distance from the origin to a point on the
coordinate system. The magnitude of avector is always positive and
is a scalar. A scalar is a quantity that does not depend on the
rotation ofthe coordinate system. Other examples of scalar
quantities are mass and temperature. A number is also ascalar.
Multiplying a vector by a scalar
When you multiply a vector by a scalar, you multiply each
component by that scalar. If a is a scalar quantity,then
a~r =< arx, ary, arz > (3)
The magnitude of the vector is thus a|~r|. Multiplying a vector
by a scalar just scales the vectorthisonly changes the magnitude of
the vector and not the direction unless the scalar is negative.
Multiplying avector by 1, reverses the vector. In other words, ~r
points in the opposite direction as ~r.
The example below shows vector ~B and the result of multiplying
it by 2 and the result of multiplying itby 1.
Figure 1: Multiplying a vector by a scalar.
Adding vectors algebraically
When adding vectors, you must add the components separately.
~A+ ~B =< (Ax +Bx), (Ay +By), (Az +Bz) > (4)
Adding vectors graphically
Vectors can also be added graphically. To add a bunch of
vectors, draw them all tail-to-head, one after theother. Then the
sum of the vectors is a vector from the tail of the first vector to
the head of the last vector.See the example below for ~A+ ~B. The
order in which you add the vectors is irrelevant, so ~A+ ~B = ~B +
~A.
Figure 2: Vector addition.
Subtracting vectors
When subtracting vector ~B from ~A, consider it as adding ~B to
~A. Its the same thing! Just reverse ~Bbefore adding it to ~A.
~A ~B = ~A+ ~B =< (Ax Bx), (Ay By), (Az Bz) > (5)
Figure 3: Vector subtraction.
When you frequently subtract one vector from another vector, you
find that theres a shortcut. Justdraw the two vectors ~A and ~B
tail to tail. Then the vector ~A ~B is the vector drawn from the
head of ~Bto the head of ~A.
Figure 4: Vector subtraction shortcut for two vectors.
NOTE: ~A ~B is in the opposite direction as ~B ~A.
Unit Vector
A unit vector has a magnitude of 1. A unit vector in the
direction of ~r is
r =~r
|~r|(6)
r = (7)
A few specially defined unit vectors are i, j, and k which point
along the x, y and z axes, respectively.They are written as
i =< 1, 0, 0 > (8)
j =< 0, 1, 0 > (9)
k =< 0, 0, 1 > (10)
Using these definitions, a vector ~r is sometimes written as
~r = rxi+ ry j + rz k (11)
Direction of a vector
If is the angle a vector makes with the +x axis, is the angle a
vector makes with the +y axis, and isthe angle a vector makes with
the +z axis, then
cos =rx|~r|
(12)
cos =ry|~r|
(13)
cos =rz|~r|
(14)
Application
1. A particle moves from the position ~r1 = (2m)i + (4m)j to ~r2
= (6m)i + (1m)j. Draw the positionvectors and determine the
displacement vector, ~r, both graphically and algebraically.
Express the dis-placement vector using unit vector notation. Also
find its magnitude and direction with respect to the +xaxis.
2. A soccer player undergoes two successive displacements, ~rA =
(15m)i + (5m)j and then ~rB =(15m)i+(10m)j. What is the total
displacement of the soccer player? Determine it both algebraically
andgraphically. Also determine the magnitude and direction of the
total displacement. The total displacementis the sum of individual
displacements along the way. Im curious, from the information in
the problem canyou determine the path the soccer player took during
these displacements?
