Question

a) Determine if the line [x,y,z] = [4,1,-2] +t[3,1,-5] has any x, y, or z-intercepts.

b) Under what conditions will a line parallel to [x,y,z] = [4,1,-2] +t[3,1,-5] have only
i. an x-intercept
ii. a y-intercept
iii. a z-intercept

c) Under what conditions (if any) will a line parallel to the one given have any x,y, or z - intercept?

Answer 1

a) It does have intercepts with x=0, y=0 and z=0 planes. In each case, a t-value exists.

b) A line parallel to
[x,y,z] = [4,1,-2] +t[3,1,-5]
would have to be of the form
[x,y,z] = [a,b,c] +t[3,1,-5]
It could not be parallel to x=0, y=0 or z=0 planes because, regardless of a,b or c, a t value could always be found to cross either of the three planes.
The answer is "none"

c) A line parallel to
[x,y,z] = [4,1,-2] +t[3,1,-5]
will always have x, y and z intercepts