Page 85 - 35Linear Algebra

4.2 Hyperplanes 85

4.2 Hyperplanes

n
Vectors in R are impossible to visualize unless n is 1,2, or 3. However,
familiar objects like lines and planes still make sense for any value of n: The
line L along the direction defined by a vector v and through a point P labeled
by a vector u can be written as

L = {u + tv | t ∈ R} .

Sometimes, since we know that a point P corresponds to a vector, we will
be lazy and just write L = {P + tv | t ∈ R}.

    
 1 1 

  
0
2
 
    4
Example 45   + t   t ∈ R describes a line in R parallel to the x 1 -axis.
0
 3  
 

  
4 0 
 
Given two non-zero vectors u, v, they will usually determine a plane,

unless both vectors are in the same line, in which case, one of the vectors
is a scalar multiple of the other. The sum of u and v corresponds to laying
the two vectors head-to-tail and drawing the connecting vector. If u and v
determine a plane, then their sum lies in the plane determined by u and v.

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