Thread: Equation of a line in R^3

1. Equation of a line in R^3

Find the equation of a line passing through points (2,4,-3) and (3,-1,1).

How would I go about doing this?

i understadn they are vectors, should I use matricies?

2. Originally Posted by VkL
Find the equation of a line passing through points (2,4,-3) and (3,-1,1).

How would I go about doing this?

i understadn they are vectors, should I use matricies?
1. Find a direction vector v$\displaystyle =<a,b,c>$

2. Use the direction numbers $\displaystyle a,b,c$ with one of your points to obtain a set of parametric equations.

3. I'm sorry, I don't mean to annoy, I do not understand.

4. Originally Posted by VkL
I'm sorry, I don't mean to annoy, I do not understand.
That's okay.

Direction vector: $\displaystyle \vec{PQ}=<3-2,-1-4,1-(-3)>=<1,-5,4>=<a,b,c>$

So, the parametric representation of a line in 3 space is

$\displaystyle x=x_1+at$, $\displaystyle y=y_1+bt$, and $\displaystyle z=z_1+ct$

Finish up. If you get stuck, show me where.

5. would it be,

x = 2 + t
y = 4 -3t
z = -3 + 4t

?

6. Originally Posted by VkL
would it be,

x = 2 + t
y = 4 -5t
z = -3 + 4t

?
Correction, but yeah.

7. thank you very much!

now if I want to find out where this line intersects the xZ-Plane, that would mean y = 0

so I put,

0 = 4 -5t

t= (4/5)

? I am not sure, do I plug the t in to the other equations?

8. vectors*

9. Originally Posted by VkL
thank you very much!

now if I want to find out where this line intersects the xZ-Plane, that would mean y = 0

so I put,

0 = 4 -5t

t= (4/5)

? I am not sure, do I plug the t in to the other equations?
Well, sure because you need the other coordinates. IE Intersection point R: $\displaystyle R(?,0,?)$

10. right, thank you very much!

11. sorry again, If I am asked whether a certain point (x, y, z) lies on the line...I plug them into the vectors adn solve for t? and the t should be the same?

12. Originally Posted by VkL
sorry again, If I am asked whether a certain point (x, y, z) lies on the line...I plug them into the vectors adn solve for t? and the t should be the same?
Indeed!

13. I keep messing up on this for some reason!

Question is "decide if the lines l1 & l2 intersect or are skew or distinct..

Just to make sure..

Distinct means the parameter t is different for vectors x y and z ?
parellel means that 1 vector has to be a scalar multipal of the other
now skew means what??

saw I find that line 1 and line 2 are not parallel
example,

L1: x = 1+t
y = -2 + 3t
z = 4 - t

L2: x = 2t
y = 3 + t
z = -3 + 4t

they are not parallel. now..

what would setting the x = x y = y z = z tell me??
setting
1+t = 2t
-2 + 3t = 3 + t
4 - t = -3 + 4t

Ok, say I solve the matrix, w.e... I get a value of t.. what does that value mean?

14. ?