# Thread: Equation of a line

1. ## Equation of a line

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:

$\frac {x-4}{-2}=\frac {y+5}{3}=\frac {z-3}{0}$ (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:

$\frac {x-4}{-2}=\frac {y+5}{3}$ (2)

z=3 (how do you get that?)

Equations 1 and 2 mean the same thing. Equation 2 is usually used.

2. The usual way to write this is $\frac{x-4}{-2}=\frac{y+5}{3};~z=3$.
The z-value is constant. There is no change in the z-direction.

3. 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.

4. Do you mind answering the questions I wrote in my first post? There is a reason I wrote them.

5. Originally Posted by chengbin
Do you mind answering the questions I wrote in my first post? There is a reason I wrote them.
What questions do you think that the two of us did not answer?
Because of this question, I wonder if you even understand the question.

6. Ahh!
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?)
$\frac{a}{0}= x$ 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 $\frac{z- 3}{0}$ to make any sense, the numerator must be 0: z- 3= 0 so z= 3.

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.

7. Originally Posted by HallsofIvy
Ahh!

$\frac{a}{0}= x$ 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 $\frac{z- 3}{0}$ to make any sense, the numerator must be 0: z- 3= 0 so z= 3.

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.
Thanks a lot for clearing this up. Just what I needed to understand this.