Form a matrix A from the 3 given vectors in the space of W.
W perpendicular is orthogonal to W, means every vector in that space is orthogonal to every vector in W.
This is possible when A X = O
Find row reduced Echelon form of A :
========================Dimension of W
= rank of A = number of independent rows in A = 2Null space
is the orthogonal space (orthogonal complement) of A, It has [ 1, 2, 4 ].
Its dimension = 1
( = n - dimension of W)
The operations performed on 1st row to become O are : 7 row1 - 4 Row2 + 2 Row3 =
7 * [2, 0, 4 ] - 4 * [ -1, 2, 0 ] + 2 [ 3, 1, 7 ]
W is a plane in in three dimensional space. x = 2 y = - 2 z
(it passes through origin)
W perpendicular is the Line perpendicular to the plane of W.