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Vector Review
Notation:
Vectors in R3 can be written as an ordered triple:

a  (a x , a y , a z )
They can be writes in terms of the unit vectors in the x, y, and z directions:

a  a x iˆ  a y ˆj  a z kˆ
Or:

a  ax xˆ  a y yˆ  az zˆ
They can also be written as a column vector (i.e. a 3 x 1 matrix):
a x 
  
a  a y 
 a z 
All of these representations of vectors are interchangeable, and depend
completely on personal preference. For the rest of this review, we will use the
ordered triple notation.
Vector addition:


Given: a  (a x , a y , a z ) and b  (bx , by , bz )
 
a  b  ( a x  b x , a y  b y , a z  bz )
Visualization of vector addition:
Scalar Multiplication:

Given: a  (a x , a y , a z )

ka  (kax , kay , kaz )
Multiplying a vector by a scalar changes the length of the vector. For example,
multiplying a vector by 2 makes the vector twice as long, but does not change the
direction:
If the scalar multiple is negative, then the multiplication changes the length of the
vector, and makes it point in the opposite direction. For example, multiplying a
vector by -2 makes the vector twice as long, and also makes the vector point in
the opposite direction:
Magnitude of a Vector:
The Magnitude of a vector is also known as the “length” or “norm” of a vector.
For vectors in R3, this is given by the Pythagorean Theorem:

a  a x2  a y2  a z2
Unit Vectors:
→ A unit vector is a vector with a magnitude of one (that is, a vector with “unit
length.”)
→ A unit vector is often (but not always) denoted with a “hat” instead of an
arrow.
→ Unit vectors can be used to signify a direction.

→ Given a vector a  (a x , a y , a z ) , we can find a unit vector that points in the

same direction as a :

a
aˆ   
|a|

ay
ax
az
 
,
,
2
2
2
2
2
2
2
2
2
2
a x  a y  a z  a x  a y  a z
ax  a y  az
a x  a y2  a z2
(a x , a y , a z )




→ We can compactly express the magnitude and direction of a vector by using
the unit vector:
 
a | a | aˆ


This reads: “the vector a has a magnitude | a | and points in the direction â .”

→ Physics notation: We will use a (no arrow, plain type) instead of | a | .

e.g. a  aaˆ
A Very Special Unit Vector:

→ The position vector r  ( x, y, z ) is a vector from the origin to an arbitrary
point in R3.
→ The magnitude of the position vector is the distance from the origin to (x,y,z).

r | r | x 2  y 2  z 2

→ The unit radial vector is a unit vector that points in the same direction as r .


r 
x
y
z

rˆ  
,
,
2
2
2
2
2
2 
r  x2  y2  z 2
x y z
x y z 

→ Example: The electric field created by a point charge q located at the origin is

q
rˆ . Try writing this in terms of x, y, and z!
given by: E 
4 0 r 2
Vector Multiplication: The Dot Product
Also known as “the scalar product” and “the inner product”


Definition: given a  (a x , a y , a z ) and b  (bx , by , bz ) ,
 
a  b  a x bx  a y b y  a z bz
Geometric properties:
   
a  b | a || b | cos( )
  
a  a | a |2

 

If a  b  0 , a is orthogonal to b
Application: Vector Projection 



Given two vectors a and b , find the vector p that is the projection of a in the

direction of b .


Since p points in the same direction as b ,

We can express p as:

  ˆ  b
p | p | b | p | 
|b |
From the picture, we can see that
 

a , p , and q all form a right triangle,

where a is the hypotenuse. Now we can find the magnitude of the vector

projection p :
 
| p || a | cos( ) , which becomes, via the definition of the dot product:
 
 a b
| p | 
|b |

To get the vector p , just multiply the magnitude by a unit vector in the same
direction, namely b̂ :
 
 
 a b   a b 
p   2 b     b
|b |
b b 
The Cross Product:
Also known as “the vector product”
Multiplies two vector to get another vector
The result of the cross product is orthogonal to both original vectors
Definition:


Given a  (a x , a y , a z ) and b  (bx , by , bz ) ,
iˆ


a  b  ax
bx
ˆj
ay
by
kˆ
a z  ( a y bz  a z b y , a z bx  a x bz , a x b y  a y bx )
bz
Properties:
   
| a  b || a || b | sin( )
  
a  (a  b )  0
  
b  (a  b )  0
 
 
a  b  (b  a )
Area of a parallelogram:


If a and b are adjacent edges of a parallelogram,
 
then | a  b | is the area of the parallelogram.
Area of a triangle:


If a and b are adjacent edges of triangle,
1  
then | a  b | is the area of the triangle.
2
The cross product obeys the “right hand rule”

  

If a is parallel to b , then a  b  0