Vector equation of a line is where is the position vector of the point a through which our line passes through and is the vector parallel to our line and is the general vector of a line satisfying these conditions and is a constant.

The direction cosines of the x-axis are (1,0,0), direction cosines are the direction ratios of a unit vector.

The equation of a line parallel to x-axis and passing through the origin is,

= (0+0+0) + λ(1+0+0)

= λ

Therefore, this is the family of lines satisfying the condition of the question.