This document provides the solution to a math problem involving vectors. It shows that the vectors (1,3,3), (1,6,6), and (1,-6,-6) are parallel to each other and equal to zero when their dot products are calculated. This means the vectors are linearly dependent and span a line through the origin in three-dimensional space rather than filling out the full three-dimensional space.

1. 1
(MMP) Problem No. 20
Solution

2. 2
2
3
4
2 1 1
6 3 3
( 1) (3 3)
( 1) (6 6)
( 1) ( 6 6) 0
i j k
X
i X
j X
k X
parallel
 
 
   
  
   

b
