Inner Product, Length and
Orthogonality
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INNER PRODUCT
For vectors u and v in ℝn we can define their INNER PRODUCT.
 
u  v  u1v1  u2v2    unvn
This is also called the “dot product”. We have been using dot products to do
matrix arithmetic, so it should be a familiar computation.
Some properties of inner products:
   
u v  v u
      
u  v  w  u  w  v  w
 
 


cu  v  cu  v  u  cv
 
uu  0
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LENGTH of a vector
For a vector v in ℝn we can define the LENGTH (or NORM) of v.

v 
 
vv 
2
2
v1
 v2



v
n
2
If v is a vector in ℝ2 you should recognize this as the length of the
hypotenuse of a right triangle (i.e. the Pythagorean Theorem)
Note that if a vector is multiplied by a constant, then its length is multiplied
by the same constant:


cv  c  v
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LENGTH of a vector
A vector with length=1 is called a UNIT VECTOR.
Any vector can be turned into a unit vector by dividing by its length.


v
u 
v
u is a vector that points the same
direction as v, but has length=1.
When we create a unit vector in this
way we say that vector v has been
“normalized”.
The DISTANCE between vectors u and v is the length of their difference:
 
 
distu, v  u  v
This should coincide with the usual “distance formula” that you know for
finding the distance between 2 points in ℝ2.
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ORTHOGONALITY
We say that vectors u and v in ℝn are ORTHOGONAL if their inner product is 0.
When two vectors are perpendicular (the angle between them is 90°) we can
also call them “orthogonal”. Just a new word for a familiar property.
u and v are orthogonal when u•v=0
Here is a formula for the angle between two vectors:
 
 
u  v  u  v  cos
Prepared by Vince Zaccone
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EXAMPLES
Given these vectors in ℝ3, find the following:
 
1) a  b

2) a

3) b
1 
   
a   2 b 
 2 


4)the angle between a and b


5) unit vectors for a and b
 3 
 0 
 
 4
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
EXAMPLES
Given these vectors in ℝ3, find the following:
 
1) a  b

2) a

3) b
1 
   
a   2 b 
 2 


4)the angle between a and b


5) unit vectors for a and b
 3 
 0 
 
 4
1) Here is the calculation:
 3 
 
 
a  b  1  2 2   0   1  3   2  0  2  4  5
 4
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Assistance Services at UCSB
EXAMPLES
Given these vectors in ℝ3, find the following:
 
1) a  b

2) a

3) b
1 
   
a   2 b 
 2 


4)the angle between a and b


5) unit vectors for a and b
 3 
 0 
 
 4
2) Here is the calculation:

a  12   22  22  9  3
Prepared by Vince Zaccone
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EXAMPLES
Given these vectors in ℝ3, find the following:
 
1) a  b

2) a

3) b
1 
   
a   2 b 
 2 


4)the angle between a and b


5) unit vectors for a and b
 3 
 0 
 
 4
3) Here is the calculation:

b  32  02   42  25  5
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB
EXAMPLES
Given these vectors in ℝ3, find the following:
 
1) a  b

2) a

3) b
1 
   
a   2 b 
 2 


4)the angle between a and b


5) unit vectors for a and b
 3 
 0 
 
 4
4) Here is the calculation:
 
 
a  b  a  b  cos
 5  3  5  cos
5
15
 cos
3 
  cos 1 1  109 .5
Prepared by Vince Zaccone
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Assistance Services at UCSB
EXAMPLES
Given these vectors in ℝ3, find the following:
 
1) a  b

2) a

3) b
1 
   
a   2 b 
 2 


4)the angle between a and b


5) unit vectors for a and b
 3 
 0 
 
 4
5) To get a unit vector, divide each
component by the length of the vector:
1 

a
23 
   3
a
2 
 3
3 

b
 5
  0 
b
 4 
 5
both of these unit vectors have
length=1 (check this!) and
point in the same directions as
the original vectors.
Prepared by Vince Zaccone
For Campus Learning
Assistance Services at UCSB