Line intersecting 2 lines and parallel to another
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The problem is :
Find parametric equations of the line L intersecting the given lines L1 and L2 and parallel to the given line L3.
L1: x = 1 + (t1), y = 2 + 2(t1), z = -2 + (t1)
L2: x = 2 + (t2), y = 1 + 2(t2), z = 3 + 3(t2)
L3: x = 1 + 2(t3), y = 1 + 7(t3), z = 1 + 3(t3)
(t1), (t2) and (t3) are in R
Thank you
linear-algebra geometry vectors 3d
New contributor
add a comment |
up vote
1
down vote
favorite
The problem is :
Find parametric equations of the line L intersecting the given lines L1 and L2 and parallel to the given line L3.
L1: x = 1 + (t1), y = 2 + 2(t1), z = -2 + (t1)
L2: x = 2 + (t2), y = 1 + 2(t2), z = 3 + 3(t2)
L3: x = 1 + 2(t3), y = 1 + 7(t3), z = 1 + 3(t3)
(t1), (t2) and (t3) are in R
Thank you
linear-algebra geometry vectors 3d
New contributor
What have you tried so far? Are you stuck at some point? or at some concept?
– Andrei
yesterday
I know that the direction vector of L is parallel to that of L3 if it is a scalar multiple of the direction vector of L3... Now how do I find the points where L intersects L1 and L2?
– L. M. E.
yesterday
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The problem is :
Find parametric equations of the line L intersecting the given lines L1 and L2 and parallel to the given line L3.
L1: x = 1 + (t1), y = 2 + 2(t1), z = -2 + (t1)
L2: x = 2 + (t2), y = 1 + 2(t2), z = 3 + 3(t2)
L3: x = 1 + 2(t3), y = 1 + 7(t3), z = 1 + 3(t3)
(t1), (t2) and (t3) are in R
Thank you
linear-algebra geometry vectors 3d
New contributor
The problem is :
Find parametric equations of the line L intersecting the given lines L1 and L2 and parallel to the given line L3.
L1: x = 1 + (t1), y = 2 + 2(t1), z = -2 + (t1)
L2: x = 2 + (t2), y = 1 + 2(t2), z = 3 + 3(t2)
L3: x = 1 + 2(t3), y = 1 + 7(t3), z = 1 + 3(t3)
(t1), (t2) and (t3) are in R
Thank you
linear-algebra geometry vectors 3d
linear-algebra geometry vectors 3d
New contributor
New contributor
New contributor
asked yesterday
L. M. E.
84
84
New contributor
New contributor
What have you tried so far? Are you stuck at some point? or at some concept?
– Andrei
yesterday
I know that the direction vector of L is parallel to that of L3 if it is a scalar multiple of the direction vector of L3... Now how do I find the points where L intersects L1 and L2?
– L. M. E.
yesterday
add a comment |
What have you tried so far? Are you stuck at some point? or at some concept?
– Andrei
yesterday
I know that the direction vector of L is parallel to that of L3 if it is a scalar multiple of the direction vector of L3... Now how do I find the points where L intersects L1 and L2?
– L. M. E.
yesterday
What have you tried so far? Are you stuck at some point? or at some concept?
– Andrei
yesterday
What have you tried so far? Are you stuck at some point? or at some concept?
– Andrei
yesterday
I know that the direction vector of L is parallel to that of L3 if it is a scalar multiple of the direction vector of L3... Now how do I find the points where L intersects L1 and L2?
– L. M. E.
yesterday
I know that the direction vector of L is parallel to that of L3 if it is a scalar multiple of the direction vector of L3... Now how do I find the points where L intersects L1 and L2?
– L. M. E.
yesterday
add a comment |
2 Answers
2
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oldest
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up vote
0
down vote
accepted
Let us use vector notation for the sake of simplicity:
$L_1: vec r_1+t_1vec l_1$ ,where $vec l_1=(1,2,1)$ and $vec r_1=(1,2,-2)$.
Likewise
$L_2: vec r_2+t_2vec l_1$ where $vec l_2=(1,2,3)$ and $vec r_2=(2,1,3)$. Let us call the line $L$ that intersects $L_1$ and $L_2$ and is parallel to $L_3$. So we get
$L: vec r+t vec l_3$ ,where $vec l_3=(2,7,3)$ and $vec r$ yet to be determined.
From intersection of $L_1$ and $L$ we get:
$$vec r_1+t_1 vec l_1=vec r+ t'_1 vec l_3$$
And from intersection of $L_2$ and $L$ we get:
$$vec r_2+t_2 vec l_2=vec r+ t'_2 vec l_3$$
If we subtract the last two vector equation we get:
$$vec r_1-vec r_2+t_1 vec l_1-t_2 vec l_2=(t'_1-t'_2)vec l_3$$
If we substitute the numerical values we get three equations as below:
$$left{begin{array}{c}-1+t_1-t_2=2(t'_1-t'_2) \1+2(t_1-t_2)=7(t'_1-t'_2)\-5+t_1-3t_2=3(t'_1-t'_2)end{array}right.$$
We can consider $t_1$ , $t_2$ and $t'_1-t'_2$ as three unknowns in the system of three equations to get: $t_1=0.5$, $t_2=-2.5$, and $t'_1-t'_2=1$.
So from intersecting $L$ and $L_1$ we can get $vec r$ as below
$$vec r=vec r_1+t_1 vec l_1-t'_1 vec l_3=(1,2,-2)+0.5(1,2,1)-t'_1 vec l_3=(1.5,3,-1.5)-t'_1 vec l_3$$
Then the parametric equation for $L$ will be:
$$L: vec r+t vec l_3=(1.5,3,-1.5)-t'_1 vec l_3+t vec l_3$$
or by having $t''=t-t'$ we get
$$L: vec r+t vec l_3=(1.5,3,-1.5)+t'' vec l_3$$
or
$$L: vec r+t vec l_3=(1.5,3,-1.5)+t'' (2,7,3)$$
or
$$ L: x=1.5+2t'', y=3+7t'', z=-1.5+3t''$$
I'm pretty sure the line L can intersect L1 at some point P1 and intersect L2 at another point P2, with P1 ≠ P2
– L. M. E.
yesterday
add a comment |
up vote
0
down vote
If two lines intersect, then they are coplanar. In particular, $L$ and $L_1$ must lie on the same plane, as must $L$ and $L_2$, therefore $L$ is the intersection of these two planes. The cross product of the direction vectors of each of these pairs of lines is normal to the plane in which they lie, and you can extract a known point on each of $L_1$ and $L_2$ from their parametric equations, so you can easily construct an equation for each of these planes. Once you have the two equations, finding their intersection is also straightforward. Indeed, you only need to find a point in the intersection since you already know what the direction vector of the line must be, but it’s worth going through the entire calculation to ensure that the planes do in fact intersect.
Specifically, we have for the first plane’s normal $(2,7,3)times(1,2,1)=(1,1,-3)$ and so an equation for it is $$x+y-3z=9.$$ For the second plane, we have the normal $(2,7,3)times(1,2,3)=(15,-3,-3)$, giving the equation $$5x-y-z=6.$$ These two normals are not parallel, so the planes do intersect in a line. I’ll leave the rest to you.
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Let us use vector notation for the sake of simplicity:
$L_1: vec r_1+t_1vec l_1$ ,where $vec l_1=(1,2,1)$ and $vec r_1=(1,2,-2)$.
Likewise
$L_2: vec r_2+t_2vec l_1$ where $vec l_2=(1,2,3)$ and $vec r_2=(2,1,3)$. Let us call the line $L$ that intersects $L_1$ and $L_2$ and is parallel to $L_3$. So we get
$L: vec r+t vec l_3$ ,where $vec l_3=(2,7,3)$ and $vec r$ yet to be determined.
From intersection of $L_1$ and $L$ we get:
$$vec r_1+t_1 vec l_1=vec r+ t'_1 vec l_3$$
And from intersection of $L_2$ and $L$ we get:
$$vec r_2+t_2 vec l_2=vec r+ t'_2 vec l_3$$
If we subtract the last two vector equation we get:
$$vec r_1-vec r_2+t_1 vec l_1-t_2 vec l_2=(t'_1-t'_2)vec l_3$$
If we substitute the numerical values we get three equations as below:
$$left{begin{array}{c}-1+t_1-t_2=2(t'_1-t'_2) \1+2(t_1-t_2)=7(t'_1-t'_2)\-5+t_1-3t_2=3(t'_1-t'_2)end{array}right.$$
We can consider $t_1$ , $t_2$ and $t'_1-t'_2$ as three unknowns in the system of three equations to get: $t_1=0.5$, $t_2=-2.5$, and $t'_1-t'_2=1$.
So from intersecting $L$ and $L_1$ we can get $vec r$ as below
$$vec r=vec r_1+t_1 vec l_1-t'_1 vec l_3=(1,2,-2)+0.5(1,2,1)-t'_1 vec l_3=(1.5,3,-1.5)-t'_1 vec l_3$$
Then the parametric equation for $L$ will be:
$$L: vec r+t vec l_3=(1.5,3,-1.5)-t'_1 vec l_3+t vec l_3$$
or by having $t''=t-t'$ we get
$$L: vec r+t vec l_3=(1.5,3,-1.5)+t'' vec l_3$$
or
$$L: vec r+t vec l_3=(1.5,3,-1.5)+t'' (2,7,3)$$
or
$$ L: x=1.5+2t'', y=3+7t'', z=-1.5+3t''$$
I'm pretty sure the line L can intersect L1 at some point P1 and intersect L2 at another point P2, with P1 ≠ P2
– L. M. E.
yesterday
add a comment |
up vote
0
down vote
accepted
Let us use vector notation for the sake of simplicity:
$L_1: vec r_1+t_1vec l_1$ ,where $vec l_1=(1,2,1)$ and $vec r_1=(1,2,-2)$.
Likewise
$L_2: vec r_2+t_2vec l_1$ where $vec l_2=(1,2,3)$ and $vec r_2=(2,1,3)$. Let us call the line $L$ that intersects $L_1$ and $L_2$ and is parallel to $L_3$. So we get
$L: vec r+t vec l_3$ ,where $vec l_3=(2,7,3)$ and $vec r$ yet to be determined.
From intersection of $L_1$ and $L$ we get:
$$vec r_1+t_1 vec l_1=vec r+ t'_1 vec l_3$$
And from intersection of $L_2$ and $L$ we get:
$$vec r_2+t_2 vec l_2=vec r+ t'_2 vec l_3$$
If we subtract the last two vector equation we get:
$$vec r_1-vec r_2+t_1 vec l_1-t_2 vec l_2=(t'_1-t'_2)vec l_3$$
If we substitute the numerical values we get three equations as below:
$$left{begin{array}{c}-1+t_1-t_2=2(t'_1-t'_2) \1+2(t_1-t_2)=7(t'_1-t'_2)\-5+t_1-3t_2=3(t'_1-t'_2)end{array}right.$$
We can consider $t_1$ , $t_2$ and $t'_1-t'_2$ as three unknowns in the system of three equations to get: $t_1=0.5$, $t_2=-2.5$, and $t'_1-t'_2=1$.
So from intersecting $L$ and $L_1$ we can get $vec r$ as below
$$vec r=vec r_1+t_1 vec l_1-t'_1 vec l_3=(1,2,-2)+0.5(1,2,1)-t'_1 vec l_3=(1.5,3,-1.5)-t'_1 vec l_3$$
Then the parametric equation for $L$ will be:
$$L: vec r+t vec l_3=(1.5,3,-1.5)-t'_1 vec l_3+t vec l_3$$
or by having $t''=t-t'$ we get
$$L: vec r+t vec l_3=(1.5,3,-1.5)+t'' vec l_3$$
or
$$L: vec r+t vec l_3=(1.5,3,-1.5)+t'' (2,7,3)$$
or
$$ L: x=1.5+2t'', y=3+7t'', z=-1.5+3t''$$
I'm pretty sure the line L can intersect L1 at some point P1 and intersect L2 at another point P2, with P1 ≠ P2
– L. M. E.
yesterday
add a comment |
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Let us use vector notation for the sake of simplicity:
$L_1: vec r_1+t_1vec l_1$ ,where $vec l_1=(1,2,1)$ and $vec r_1=(1,2,-2)$.
Likewise
$L_2: vec r_2+t_2vec l_1$ where $vec l_2=(1,2,3)$ and $vec r_2=(2,1,3)$. Let us call the line $L$ that intersects $L_1$ and $L_2$ and is parallel to $L_3$. So we get
$L: vec r+t vec l_3$ ,where $vec l_3=(2,7,3)$ and $vec r$ yet to be determined.
From intersection of $L_1$ and $L$ we get:
$$vec r_1+t_1 vec l_1=vec r+ t'_1 vec l_3$$
And from intersection of $L_2$ and $L$ we get:
$$vec r_2+t_2 vec l_2=vec r+ t'_2 vec l_3$$
If we subtract the last two vector equation we get:
$$vec r_1-vec r_2+t_1 vec l_1-t_2 vec l_2=(t'_1-t'_2)vec l_3$$
If we substitute the numerical values we get three equations as below:
$$left{begin{array}{c}-1+t_1-t_2=2(t'_1-t'_2) \1+2(t_1-t_2)=7(t'_1-t'_2)\-5+t_1-3t_2=3(t'_1-t'_2)end{array}right.$$
We can consider $t_1$ , $t_2$ and $t'_1-t'_2$ as three unknowns in the system of three equations to get: $t_1=0.5$, $t_2=-2.5$, and $t'_1-t'_2=1$.
So from intersecting $L$ and $L_1$ we can get $vec r$ as below
$$vec r=vec r_1+t_1 vec l_1-t'_1 vec l_3=(1,2,-2)+0.5(1,2,1)-t'_1 vec l_3=(1.5,3,-1.5)-t'_1 vec l_3$$
Then the parametric equation for $L$ will be:
$$L: vec r+t vec l_3=(1.5,3,-1.5)-t'_1 vec l_3+t vec l_3$$
or by having $t''=t-t'$ we get
$$L: vec r+t vec l_3=(1.5,3,-1.5)+t'' vec l_3$$
or
$$L: vec r+t vec l_3=(1.5,3,-1.5)+t'' (2,7,3)$$
or
$$ L: x=1.5+2t'', y=3+7t'', z=-1.5+3t''$$
Let us use vector notation for the sake of simplicity:
$L_1: vec r_1+t_1vec l_1$ ,where $vec l_1=(1,2,1)$ and $vec r_1=(1,2,-2)$.
Likewise
$L_2: vec r_2+t_2vec l_1$ where $vec l_2=(1,2,3)$ and $vec r_2=(2,1,3)$. Let us call the line $L$ that intersects $L_1$ and $L_2$ and is parallel to $L_3$. So we get
$L: vec r+t vec l_3$ ,where $vec l_3=(2,7,3)$ and $vec r$ yet to be determined.
From intersection of $L_1$ and $L$ we get:
$$vec r_1+t_1 vec l_1=vec r+ t'_1 vec l_3$$
And from intersection of $L_2$ and $L$ we get:
$$vec r_2+t_2 vec l_2=vec r+ t'_2 vec l_3$$
If we subtract the last two vector equation we get:
$$vec r_1-vec r_2+t_1 vec l_1-t_2 vec l_2=(t'_1-t'_2)vec l_3$$
If we substitute the numerical values we get three equations as below:
$$left{begin{array}{c}-1+t_1-t_2=2(t'_1-t'_2) \1+2(t_1-t_2)=7(t'_1-t'_2)\-5+t_1-3t_2=3(t'_1-t'_2)end{array}right.$$
We can consider $t_1$ , $t_2$ and $t'_1-t'_2$ as three unknowns in the system of three equations to get: $t_1=0.5$, $t_2=-2.5$, and $t'_1-t'_2=1$.
So from intersecting $L$ and $L_1$ we can get $vec r$ as below
$$vec r=vec r_1+t_1 vec l_1-t'_1 vec l_3=(1,2,-2)+0.5(1,2,1)-t'_1 vec l_3=(1.5,3,-1.5)-t'_1 vec l_3$$
Then the parametric equation for $L$ will be:
$$L: vec r+t vec l_3=(1.5,3,-1.5)-t'_1 vec l_3+t vec l_3$$
or by having $t''=t-t'$ we get
$$L: vec r+t vec l_3=(1.5,3,-1.5)+t'' vec l_3$$
or
$$L: vec r+t vec l_3=(1.5,3,-1.5)+t'' (2,7,3)$$
or
$$ L: x=1.5+2t'', y=3+7t'', z=-1.5+3t''$$
edited yesterday
answered yesterday
Arash Rashidi
558
558
I'm pretty sure the line L can intersect L1 at some point P1 and intersect L2 at another point P2, with P1 ≠ P2
– L. M. E.
yesterday
add a comment |
I'm pretty sure the line L can intersect L1 at some point P1 and intersect L2 at another point P2, with P1 ≠ P2
– L. M. E.
yesterday
I'm pretty sure the line L can intersect L1 at some point P1 and intersect L2 at another point P2, with P1 ≠ P2
– L. M. E.
yesterday
I'm pretty sure the line L can intersect L1 at some point P1 and intersect L2 at another point P2, with P1 ≠ P2
– L. M. E.
yesterday
add a comment |
up vote
0
down vote
If two lines intersect, then they are coplanar. In particular, $L$ and $L_1$ must lie on the same plane, as must $L$ and $L_2$, therefore $L$ is the intersection of these two planes. The cross product of the direction vectors of each of these pairs of lines is normal to the plane in which they lie, and you can extract a known point on each of $L_1$ and $L_2$ from their parametric equations, so you can easily construct an equation for each of these planes. Once you have the two equations, finding their intersection is also straightforward. Indeed, you only need to find a point in the intersection since you already know what the direction vector of the line must be, but it’s worth going through the entire calculation to ensure that the planes do in fact intersect.
Specifically, we have for the first plane’s normal $(2,7,3)times(1,2,1)=(1,1,-3)$ and so an equation for it is $$x+y-3z=9.$$ For the second plane, we have the normal $(2,7,3)times(1,2,3)=(15,-3,-3)$, giving the equation $$5x-y-z=6.$$ These two normals are not parallel, so the planes do intersect in a line. I’ll leave the rest to you.
add a comment |
up vote
0
down vote
If two lines intersect, then they are coplanar. In particular, $L$ and $L_1$ must lie on the same plane, as must $L$ and $L_2$, therefore $L$ is the intersection of these two planes. The cross product of the direction vectors of each of these pairs of lines is normal to the plane in which they lie, and you can extract a known point on each of $L_1$ and $L_2$ from their parametric equations, so you can easily construct an equation for each of these planes. Once you have the two equations, finding their intersection is also straightforward. Indeed, you only need to find a point in the intersection since you already know what the direction vector of the line must be, but it’s worth going through the entire calculation to ensure that the planes do in fact intersect.
Specifically, we have for the first plane’s normal $(2,7,3)times(1,2,1)=(1,1,-3)$ and so an equation for it is $$x+y-3z=9.$$ For the second plane, we have the normal $(2,7,3)times(1,2,3)=(15,-3,-3)$, giving the equation $$5x-y-z=6.$$ These two normals are not parallel, so the planes do intersect in a line. I’ll leave the rest to you.
add a comment |
up vote
0
down vote
up vote
0
down vote
If two lines intersect, then they are coplanar. In particular, $L$ and $L_1$ must lie on the same plane, as must $L$ and $L_2$, therefore $L$ is the intersection of these two planes. The cross product of the direction vectors of each of these pairs of lines is normal to the plane in which they lie, and you can extract a known point on each of $L_1$ and $L_2$ from their parametric equations, so you can easily construct an equation for each of these planes. Once you have the two equations, finding their intersection is also straightforward. Indeed, you only need to find a point in the intersection since you already know what the direction vector of the line must be, but it’s worth going through the entire calculation to ensure that the planes do in fact intersect.
Specifically, we have for the first plane’s normal $(2,7,3)times(1,2,1)=(1,1,-3)$ and so an equation for it is $$x+y-3z=9.$$ For the second plane, we have the normal $(2,7,3)times(1,2,3)=(15,-3,-3)$, giving the equation $$5x-y-z=6.$$ These two normals are not parallel, so the planes do intersect in a line. I’ll leave the rest to you.
If two lines intersect, then they are coplanar. In particular, $L$ and $L_1$ must lie on the same plane, as must $L$ and $L_2$, therefore $L$ is the intersection of these two planes. The cross product of the direction vectors of each of these pairs of lines is normal to the plane in which they lie, and you can extract a known point on each of $L_1$ and $L_2$ from their parametric equations, so you can easily construct an equation for each of these planes. Once you have the two equations, finding their intersection is also straightforward. Indeed, you only need to find a point in the intersection since you already know what the direction vector of the line must be, but it’s worth going through the entire calculation to ensure that the planes do in fact intersect.
Specifically, we have for the first plane’s normal $(2,7,3)times(1,2,1)=(1,1,-3)$ and so an equation for it is $$x+y-3z=9.$$ For the second plane, we have the normal $(2,7,3)times(1,2,3)=(15,-3,-3)$, giving the equation $$5x-y-z=6.$$ These two normals are not parallel, so the planes do intersect in a line. I’ll leave the rest to you.
edited 7 hours ago
answered yesterday
amd
28.4k21048
28.4k21048
add a comment |
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L. M. E. is a new contributor. Be nice, and check out our Code of Conduct.
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What have you tried so far? Are you stuck at some point? or at some concept?
– Andrei
yesterday
I know that the direction vector of L is parallel to that of L3 if it is a scalar multiple of the direction vector of L3... Now how do I find the points where L intersects L1 and L2?
– L. M. E.
yesterday