Vector parametric equation of line

[songoku]

Homework Statement

Let L be the line given by the equations x + y = 1 and x + 2y + z = 3. Write a vector parametric equation for L

Relevant Equations

Parametric equation:
$x=x_0 + ta$
$y=y_0+tb$
$z=z_0+tc$

I can imagine x + y = 1 to be line in xy - plane but how can x + 2y + z = 3 be a line, not a plane?

Thanks

 

[Orodruin]

Homework Statement: Let L be the line given by the equations x + y = 1 and x + 2y + z = 3. Write a vector parametric equation for L
Relevant Equations: Parametric equation:
$x=x_0 + ta$
$y=y_0+tb$
$z=z_0+tc$

I can imagine x + y = 1 to be line in xy - plane but how can x + 2y + z = 3 be a line, not a plane?

Thanks

Neither is a line. They are both planes in $\mathbb R^3$.

Two planes that are not parallel intersect each other in a line.

 

[songoku]

Neither is a line. They are both planes in $\mathbb R^3$.

Two planes that are not parallel intersect each other in a line.

Ok so it means the question is wrong to call those two equations as lines. Basically the question gives two equations of plane and asks for intersection of the two planes.

I understand the question now. Thank you very much Orodruin

 

[Orodruin]

Ok so it means the question is wrong to call those two equations as lines.

The question does not call those equations lines. The question specifies one line as the set of points satisfying both equations.

 

[FactChecker]

The points that satisfy both equations is a line. The problem statement is correct.

Suppose you set $x=t,\ \ t \in \mathbb R$.
Then from the first equation, you have $y=1-x = 1-t,\ \ t \in \mathbb R$.
Can you use the first equation to convert the second equation into an equation for $z$?

 

[Orodruin]

The points that satisfy both equations is a line. The problem statement is correct.

Huh? I never claimed anything else. In fact, it is just a repetition of what I just said …

 

[FactChecker]

Huh? I never claimed anything else. In fact, it is just a repetition of what I just said …

Sorry. Somehow I quoted the wrong post and didn't notice that I was responding to the wrong text. I must need more coffee. I'm going to fix that post.

 

[fresh_42]

Homework Statement: Let L be the line given by the equations x + y = 1 and x + 2y + z = 3. Write a vector parametric equation for L
Relevant Equations: Parametric equation:
$x=x_0 + ta$
$y=y_0+tb$
$z=z_0+tc$

I can imagine x + y = 1 to be line in xy - plane but how can x + 2y + z = 3 be a line, not a plane?

Thanks

If you have two points on a line, then you can take one as the starting point, and the difference of two as the direction to travel along by the parameter. That gives you a natural parameterization.

 

[songoku]

The question does not call those equations lines. The question specifies one line as the set of points satisfying both equations.

It seemed I interpreted it too literally. "Let L be the line given by the equations x + y = 1 and x + 2y + z = 3" in my interpretation meant the given equation are lines, which is actually not what it means by the question.

The points that satisfy both equations is a line. The problem statement is correct.

Suppose you set $x=t,\ \ t \in \mathbb R$.
Then from the first equation, you have $y=1-x = 1-t,\ \ t \in \mathbb R$.
Can you use the first equation to convert the second equation into an equation for $z$?

Yes I can

Thank you very much Orodruin, FactChecker, fresh_42

 

[Orodruin]

which is actually not what it means by the question.

Which is actually not what it means at all. The question talks about the line given by two expressions. You have simply misread the question.