MAT 2401 Discovery Lab 5.1
Objectives
 To explore the formulas of computing length of vectors and distances between
vectors in Rn .
 To investigate the properties of vectors in Rn .
 This is a preview of the early materials in MAT 2228 Multivariable Calculus. Many
of these definitions and properties can be extended to general vector spaces
(5.2-5.3).
Instructions
 Do not look up any references including the textbook and internet.
 Use correct notations and do not skip steps.
 Two persons per group.
 Do not communicate with other groups.
Length of a Vector in R2
1. Let v   2,1  R2 . Denote its length by v . This is also called the norm of v .
(a) Draw the vector v . Make sure you label the vector v , its length v on the diagram.
Compute its norm v by using the Pythagorean Theorem.
v 
Diagram of v   2,1  R2 .
(b) Based on your results from (a), suggest a possible formula for the length v of a
general vector v   v1 , v2   R2 .
v 
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Length of a Vector in Rn
The length (norm) of a vector v   v1 , v2 ,
, vn   Rn is given by
v  v12  v22 
 vn2 .
2. Compute the norm for each of the following vectors in R 3 . Simplify your answers.
(a) v  1, 1,0  .
v 
(b) v   2, 2,0  .
v 
(c) v   3, 3,0  .
v 
(d) v   3,3,0  .
v 
(e) Based on your results from (a)-(d), suggest a possible relationship between cv and
v of a general vector v  R n , and c  R .
cv 
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3. For each of the vector v in problem 2 (a)-(c), compute and simplify the vector
v
. Also, compute the length of u .
u
v
(a) v  1, 1,0  .
u
v

v
u 
(b) v   2, 2,0  .
u
v

v
u 
(c) v   3, 3,0  .
u
v

v
u 
(d) Let v  R n , and u 
v
. Suggest a possible formula for u .
v
u 
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Unit Vector in the Direction of v
v
has length 1 and has the
v
same direction as v. This vector u is called the unit vector in the direction of v.
If v   v1 , v2 ,
, vn   Rn is nonzero, then the vector u 
Distance Between Two Vectors in R2
4. Let u   u1 , u2  , v   v1 , v2   R2 .
It makes sense to define the distance between the
two vectors, d  u, v  , as the distance between the two points
d  u, v  
u1  v1   u2  v2 
2
u1, u2  ,  v1, v2  .
2
(a) Compute the vector u  v and its norm u  v .
u v 
u v 
(b) Let u   u1 , u2 ,
, un  , v   v1, v2 ,
, vn   Rn . Suggest a definition for d  u, v  .
Also suggest a possible relationship between d  u, v  and
u v .
d  u, v  
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5. Determine if each the following statement is true or false.
answers.
(a)
Explain/Justify your
d  u, v   0 for some u, v  R n .
(b) d  u, v   0
if and only if u  v .
(c) There are some u, v  R n such that d  u, v   d  v, u  .
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Dot Product in R2
6. Let u   u1 , u2  , v   v1 , v2   R2 . The dot product of the two vectors is given by
u  v  u1v1  u2v2 .
(a) Is the dot product u  v a vector or a scalar?
(b) Let u  1, 2 , v   2,1  R2 . Draw the vectors u and v. Make sure you label the
vectors. Compute the dot product u  v .
u  v  1, 2    2,1

Diagram of u  1, 2 , v   2,1  R2 .
(c) Let u  1, 2 , v   4, 2  R2 . Draw the vectors u and v. Make sure you label the
vectors. Compute the dot product u  v .
u  v  1, 2    4, 2 

Diagram of u  1, 2 , v   4, 2  R2 .
(d) Guess a geometric condition for which u  v  0.
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7. (a) Mimic the definition in problem 6, suggest a definition for the dot product in Rn .
Make sure you define all the variables you use in your definition.
(b) Two vectors in Rn are called orthogonal if u  v  0.
a non-zero vector v  R 3 such that u and v are orthogonal.
computing u  v.
Let u  1,3, 2  R3 . Find
Justify your answer by
v
uv 
(c) Let v   v1 , v2 , , vn   Rn . Show that v  v  v .
Partial solutions are given. Fill in the missing steps.
2

vv  


 

 
 




Take the dot product.




 v
Fill in the vector form of v.



2
2
Make it into a square.
Check that this is actually true.
Note that many other properties of dot product can be proved in a similar fashion.
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