The usual way to write this is .
The z-value is constant. There is no change in the z-direction.
I don't get the explanation.
Given that line k passes through point A(4,-5,3) and has direction vector (-2,3,0), the equation of line k can be expressed as:
(1)
If only the denominator is 0, then equation 1 is meaningless. (What does that mean)
If when the denominator is 0, we assume that the numerator is also 0 (why?) and the fraction can be any value (why?), then equation 1 can be expressed as:
(2)
z=3 (how do you get that?)
Equations 1 and 2 mean the same thing. Equation 2 is usually used.
You can also write that as "parametric equations":
x= 4- 2t, y= -5+ 3t, z= 3+ 0t= 3
Solving the first two equations for t, t= (4- x)/2 and t= (5+y)/3. Set those equal: (4-x)/2= (5+y)/2. Since there is no t in the z equation, you cannot solve for t- but you still have z= 3.
Ahh!
is the same as a= 0(x). But 0 times any number is 0 so that only makes sense if a= 0- that is, if "the numerator is also 0". but in that case 0= 0(x) for any x. That is why "the fraction can be any value". So for to make any sense, the numerator must be 0: z- 3= 0 so z= 3.If when the denominator is 0, we assume that the numerator is also 0 (why?) and the fraction can be any value (why?)
z=3 (how do you get that?)
That's abusing the notation slightly. In fact, you can't divide by 0. All of this should be done in terms of limits to be valid.