
B
Scalar product (dot product)


A
   
A  B | A || B | cos   Ax  Bx  Ay  B y  Az  Bz
Properties of the dot product
   
A B  B  A
  
   
A  (B  C)  A  B  A  C
 
2
A  A A


Ax  A  iˆ
Ay  A  ˆj

Az  A  kˆ
B|| A  B cos 
A||B  A cos 
B A  B sin 
A B  A sin 
 
A  B  A||B B  AB|| A
 
A B
cos    
| A || B |
Angle between vectors
   
A  B | A || B | cos 

Question1:
Question2:
 
What is A  A equal to?
Two vectors are given by
( A is the magnitude of A )
A  1.00 î  1.00 ĵ
B  1.00 î  1.00 ĵ .
1) 0
2) A
2
The angle between A and B
is ___ degrees.
3) A1/2
4) A
1. 0
2. 45
3. 90
4. 135
Example: Which pair of vectors will have the largest value for A·B?
A
A
B
Aprojected onto B
A·B = 0
More visual:
B
A
A
60° B
C
30°
B
Aprojected onto B
Aprojected onto B
A·B = AB cos60°
A·B = AB cos30°
A·B = B Aprojected onto B
Vector product (cross product)
 
A  B  AB sin 
  
A B  A
 
A  B  A B B  AB A
Right-hand rule
A×B
B
A
  
A B  C
Properties of vector product
 A  B is a vector! (A  B is a scalar)

z k
 A  B  B  A

j
 A A  0
 iˆ  iˆ  0,
jˆ  jˆ  0,
kˆ  kˆ  0
iˆ  jˆ  kˆ,
jˆ  kˆ  iˆ,
kˆ  iˆ  jˆ
y
x

i
Right-handed
coordinate
systems
A  B  (Ay Bz  Az By )iˆ  (Az Bx  Ax Bz ) jˆ  (Ax By  Ay Bx )kˆ
iˆ
 Ax
Bx
jˆ
Ay
By
kˆ
Az
Bz
 
A B
sin    
| A || B |
Example: Vectors A, B and C are on the plane of the screen.
They are drawn to scale. Compare the magnitude of these
two cross products:
A. |A×B| > |A×C|
B. |A×B| = |A×C|
C. |A×B| < |A×C|
|A×B| = AB sinθ= AC = |A×C|
C
Bcosθ
θ
And they both point out of the screen.
A
The cross product selects the part of B
that is perpendicular to the direction of A.
B
Bsinθ
Question1 : Right-handed Cartesian coordinate system:
What is the direction
of the +z axis?
What is the direction
of the +x axis?
1. Into page
2. Out of page
Question 3:
Consider two nonzero vectors A and B
with an angle  between them.
Question 4:
Two vectors have the properties that
A  B  1.25 m 2
| A  B |  1.25 m 2 .
A  (A  B)  ____?
1. A 2 Bsin  cos 
2
2. A B
3. ABsin  cos 
4. zero
The angle between A and B
is ____ degrees.
1. 45
2. 90
3. 135
4. 180
Question 5:
Question 6:
Two vectors are:
A  1.00 î  2.00 ĵ
B  3.00 î  4.00 ĵ
 
A B - ?
Then A  B  ____ .
1. 0
2. 4.5
3. 2.25
4. 2.25m 2
1.
10 k̂
2.  10 k̂
Question 7:
3. 3 î  8 ĵ
Two vectors are given by
4. 2 î  2 ĵ
A  3.00 î  4.00 ĵ
B  3.00 î  4.00 ĵ .
The angle between A and B
is ___ degrees.
1.
2.
3.
4.
106
111
116
123