Vectors Outline
Objective: Understand vector, vector components, scalar, magnitude, scalar multiplication, direction,
adding vectors (algebraically and graphically), subtracting vectors (algebraically and graphically), unit vectors.
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 a
position vector is
q
|~r| = (rx2 + ry2 + rz2 )
(2)
and represents the distance from the origin to a point on the coordinate system. The magnitude of a
vector is always positive and is a scalar. A scalar is a quantity that does not depend on the rotation of
the coordinate system. Other examples of scalar quantities are mass and temperature. A number is also a
scalar.
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 vector–this
only changes the magnitude of the vector and not the direction unless the scalar is negative. Multiplying a
vector by −1, “reverses” the vector. In other words, −~r points in the opposite direction as ~r.
~ and the result of multiplying it by 2 and the result of multiplying it
The example below shows vector B
by −1.
Figure 1: Multiplying a vector by a scalar.
Adding vectors algebraically
When adding vectors, you must add the components separately.
~+B
~ =< (Ax + Bx ), (Ay + By ), (Az + Bz ) >
A
(4)
Adding vectors graphically
Vectors can also be added graphically. To add a bunch of vectors, draw them all tail-to-head, one after the
other. Then the sum of the vectors is a vector from the tail of the first vector to the head of the last vector.
~ + B.
~ The order in which you add the vectors is irrelevant, so A
~+B
~ =B
~ + A.
~
See the example below for A
Figure 2: Vector addition.
Subtracting vectors
~ from A,
~ consider it as adding −B
~ to A.
~ It’s the same thing! Just reverse B
~
When subtracting vector B
~
before adding it to A.
~−B
~ =A
~ + −B
~ =< (Ax − Bx ), (Ay − By ), (Az − Bz ) >
A
(5)
Figure 3: Vector subtraction.
When you frequently subtract one vector from another vector, you find that there’s a shortcut. Just
~ and B
~ tail to tail. Then the vector A
~−B
~ is the vector drawn from the head of B
~
draw the two vectors A
~
to the head of A.
Figure 4: Vector subtraction shortcut for two vectors.
~−B
~ is in the opposite direction as B
~ − A.
~
NOTE: A
Unit Vector
A unit vector has a magnitude of 1. A unit vector in the direction of ~r is
r̂ =
rx
r̂ =< q
rx2
+
ry2
+
rz2
~r
|~r|
rz
rz
,q
,q
>
2
2
2
2
rx + ry + rz
rx + ry2 + rz2
(6)
(7)
A few specially defined unit vectors are î, ĵ, and k̂ which point along the x, y and z axes, respectively.
They are written as
î =< 1, 0, 0 >
(8)
ĵ =< 0, 1, 0 >
(9)
k̂ =< 0, 0, 1 >
(10)
Using these definitions, a vector ~r is sometimes written as
~r = rx î + ry ĵ + rz k̂
(11)
Direction of a vector
If α is the angle a vector makes with the +x axis, β is the angle a vector makes with the +y axis, and γ is
the angle a vector makes with the +z axis, then
rx
|~r|
ry
cos β =
|~r|
rz
cos γ =
|~r|
cos α =
(12)
(13)
(14)
Application
1. A particle moves from the position ~r1 = (2m)î + (−4m)ĵ to ~r2 = (−6m)î + (1m)ĵ. Draw the position
vectors and determine the displacement vector, ∆~r, both graphically and algebraically. Express the displacement vector using unit vector notation. Also find its magnitude and direction with respect to the +x
axis.
2. A soccer player undergoes two successive displacements, ∆~rA = (15m)î + (5m)ĵ and then ∆~rB =
(−15m)î + (10m)ĵ. What is the total displacement of the soccer player? Determine it both algebraically and
graphically. Also determine the magnitude and direction of the total displacement. The total displacement
is the sum of individual displacements along the way. I’m curious, from the information in the problem can
you determine the path the soccer player took during these displacements?