2. VECTORS
2.1 SCALARS AND VECTORS
SCALAR
- physical quantity that is completely described only by a real number
VECTOR
(e.g., time, mass, pressure, density, temperature)
- physical quantity that is completely described by both a real number
(magnitude) and a direction
(e.g., position of a point in space, displacement, force, velocity)
SYMBOLS
|U| - to denote the magnitude
U - to denote the vector (boldfaced letters)
 - an arrow for graphical representation
2.2 RULES FOR MANIPULATING VECTORS
VECTOR ADDITION
The sum of two vectors U and V is another vector W
UV  W
- TRIANGLE RULE
(if the vectors are placed head to tail, the sum is the vector from
the tail of the first vector to the head of the second/last vector)
- PARALELOGRAM RULE (consequence of the fact that the sum does not depend on the
order in which the vectors are placed head to tail)
UV  VU
- commutative
(U  V)  W  U  (V  W) - associative
If the sum is zero, vectors form a closed polygon.
PRODUCT OF A SCALAR AND A VECTOR
Product of a scalar (real number) a and a vector U is a vector aU; its magnitude is
|a||U|, and its direction is the same as the direction of U when a is positive, and opposite
to the direction of U when a is negative
Division of a vector by a scalar is defined to be a product:
U 1
 U
a a
a(bU)  (ab)U
- associative with respect to scalar multiplication
(a  b)U  aU  bU
- distributive with respect to scalar addition
a(U  V)  aU  aV
- distributive with respect to vector addition
VECTOR SUBTRACTION
The difference of two vectors U and V is obtained by adding U to the vector (1)V
U  V  U  (1)V
[V and (1)V have the opposite directions]
VECTOR COMPONENTS - a set of vectors whose sum represents the vector considered
2.3 CARTESIAN COMPONENTS IN TWO DIMENSIONS
U  U x  U y  U x i U y j
Ux and Uy are the scalar components of U
| U | U x2  U y2
Uy
U


Ux

U x | U | cos | U | sen 
U y | U | sen | U | cos 
MANIPULATING VECTORS IN TERMS OF COMPONENTS
U  V  (U x i  U y j)  (Vx i  V y j)
 (U x i  Vx i)  (U y j  V y j)
aU  a(U x i  U y j)  aU x i  aU y j
2.4 CARTESIAN COMPONENTS IN THREE DIMENSIONS
U  U x  U y  U z  U x i U y j U z k
Ux, Uy and Uz are the scalar components of U
| U | U x2  U y2  U z 2
U x | U | cos x
U y | U | cos y
U z | U | cos z
cos2 x  cos2 y  cos2 z  1
POSITION VECTOR COMPONENTS
2D
3D
rAB  (xB  xA)i  ( yB  yA) j
rAB  (xB  xA)i  ( yB  yA) j  (zB  zA)k
PRODUCTS OF VECTORS
2.5 DOT PRODUCT (uses: to determine the angle between two lines in space;
the components of a vector, parallel and normal to a given line)
U  V  | U || V | cos
The dot product of U and V is defined to be the product of the
magnitude of U, the magnitude of V, and the cosine of the angle
 between U and V when they are placed tail to tail
- The result of the dot product is a scalar
(sometimes, the dot product is called the scalar product)
- The dot product of two nonzero vectors is equal to zero if and only if the vectors
are perpendicular
PROPERTIES
U  V  V U
commutative
a (U  V)  (a U) V  U  (aV) associative with respect to scalar multiplication
U  (V  W)  U V  U W
distributive with respect to vector addition
DOT PRODUCT IN TERMS OF COMPONENTS
U  V  (U x i  U y j  U z k )  (V x i  V y j  V z k )
i  i | i || i | cos(0)  1
i  j | i || j | cos(90)  0
 U x Vx  U y V y  U z Vz
U x Vx  U y V y  U z Vz
UV
cos 

| U || V |
| U || V |
ii 1
j i  0
k i  0
ij 0
j j  1
kj0
i k  0
jk  0
k k 1
VECTOR COMPONENTS PARALLEL AND NORMAL TO A LINE
| U p | | U | cos
parallel component (projection of the vector onto the line)
e  U  | e || U | cos  | U | cos
e – unit vector parallel to the line
| Up | e  U
U p  (e  U) e
Un  U  Up
normal component
2.6 CROSS PRODUCT
U  V  | U || V | sin  e
(uses: to determine the rate of rotation of a fluid particle; to
calculate the force exerted on a charge particle by a magnetic field)
 – angle between U and V when they are placed tail to tail
e – unit vector perpendicular to both
(U, V and e – right-handed system)
- The result of the cross product is a vector
(sometimes, the cross product is called the vector product)
- The cross product of two nonzero vectors is equal to zero if and only if the vectors
are parallel
PROPERTIES
U  V   V U
not commutative
a (U  V )  (a U)  V  U  (aV )
associative with respect to scalar multiplication
U  (V  W)  U  V  U  W
distributive with respect to vector addition
CROSS PRODUCT IN TERMS OF COMPONENTS
U  V  (U x i  U y j  U z k )  (Vx i  V y j  Vz k )
i  i | i || i | sin( 0)  0
i
j
k
 Ux Uy Uz
Vx
Vy
i  j | i || j | sin( 90) e  k
Vz
2.6 MIXED TRIPLE PRODUCT
ii  0
j  i  k
k i  j
i  k  j
j k  i
k k  0
(uses: to determine the moment of a force about a line)
i
U  (V  W )  (U x i  U y j  U z k )  Vx
j
Vy
Ux U y Uz
k
Vz  Vx V y Vz
Wx W y Wz
U  (V  W)   W  (V  U )
i j  k
j j  0
k  j  i
Wx W y
Wz
interchanging any two vectors changes only the sign