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# Prove an if and only if condition for two lines to intersect in Rn

1. Let and be two lines in . Prove that they intersect if and only if is in the linear span of and .
2. Consider the two lines in ,

Determine whether they intersect.

1. Proof. Assume and intersect. Since and we know there exists an such that . This implies

This implies is in the linear span of .

Conversely, assume is in the linear span of . Then there exist such that

Thus, there is some point in both and so they intersect

2. The two given lines do not intersect. We know from part (a) that two lines and intersect if and only if is in the linear span of . In this case we have

and , . For to be in the linear span of we must have such that

But the second equation implies . The third equation would then require which gives . Then, . But then from the first equation

Hence, there are no such so is not in the linear span of . Hence, these two lines do not intersect.

# Prove that two lines in R3 intersect and determine the points of intersection

Let

Consider the lines and where is the line through parallel to and is the line through parallel to . Prove that these two lines intersect and determine the point of intersection.

Proof. We have

Then if and intersect we must have such that

From the second equation we have which implies . Plugging this into the first equation we then have which implies . Therefore, . Then we check these values of satisfy the third equation,

So, the point of intersection is

# Determine all subsets of a given set of points which lie on the same line

Consider the set of points

The points lie on a line. Determine all other subsets of three or more points which lie on the same line.

First, we notice that the third coordinate of all of the given points is 1. So, if we check that the coordinates are on the same line, then the points will be on the same line. Next, we look at which other points are on the same line as . This line is given by

Checking the remaining points we have
The point is on since implies and . Thus, for .
The point is not on since implies , but then .
The point is on since implies and then . Thus, for so is on .
The point is not on since implies , but then . SO is not on .
The point is not on since implies , but then . So is not on .

Therefore, we have that are all on the same line (hence, every subset of these with three or more elements is a subset with three or more points on the same line).

None of the other points are on the same line since is not on the unique line containing .

# Determine if a given set of points all lie on the same line

In each of the following cases determine if the given set of points all lie on the same line.

1. , , .
2. , , .
3. , , .

1. The points are not on the same line since the unique line containing is given by

But, cannot be on this line since for any , the third coordinate of every point on the line will be 1, while the third coordinate of is .

2. The points are not on the same line since the unique line containing the points is given by

But, is not on this line since if then we must have . However, with we have .

3. The points are not on the same line since the unique line containing the points is given by

But then is not on this line since if then , but then . Hence, is not on the line.

# Determine which points are on a line containing (-3,1,1) and (1,2,7)

Let be the line containing the points and . Determine which of the following points are also on .

1. ;
2. ;
3. ;
4. ;
5. ;
6. ;
7. .

First, the line containing the points and is the set of points

1. The point is not on since

But then . Hence, there is no such that , so is not on .

2. The point is on since

And then and . Hence, where . Thus, is on .

3. The point is not on since

But then . Hence, there is no such that , so is not on .

4. The point is not on since

But then . Hence, there is no such that , so is not on .

5. The point is on since

And then and . Hence, where . Thus, is on .

6. The point is on since

And then and . Hence, where . Thus, is on .

7. The point is not on since

But then . Hence, there is no such that , so is not on .

# Determine which points are on a line containing the point (-3,1,1) and parallel to the vector (1,-2,3)

Let be a point in and let be a line containing and parallel to the vector . Determine which of the following points are also on .

1. ;
2. ;
3. ;
4. ;
5. .

Given a point and a vector the line containing in the direction of is given by

1. If were on then we must have some such that

From the first equation, this requires , but then neither of the other two equations can hold. Hence, there is no such that so is not on the line.

2. If were on then we must have some such that

From the first equation, this requires , but then neither of the other two equations can hold. Hence, there is no such that so is not on the line.

3. If is on then we must have some such that

From the first equation we have . This value of also satisfies the other two equations. Hence, for . Therefore, is on the line.

4. If is on then we must have some such that

From the first equation we have . This value of also satisfies the other two equations. Hence, for . Therefore, is on the line.

5. If is on then we must have some such that

From the first equation we have . This value of also satisfies the other two equations. Hence, for . Therefore, is on the line.

# Determine which points are on a line containing (2,-1) and (-4,2)

Let be a line containing the points and . Determine which of the following points are also on .

1. ;
2. ;
3. ;
4. ;
5. ;

First, we have , so the line containing the points and is the set

1. The point is on since

Thus, for , so is on .

2. The point is not on since

Thus, for any , so is not on .

3. The point is not on since

Thus, for any , so is not on .

4. The point is not on since

Thus, for any , so is not on .

5. The point is on since

Thus, for , so is on .

# Determine which points are on a given line

Let be a line containing the points and . Determine which of the following points are also on .

1. ;
2. ;
3. ;
4. ;
5. ;

By Theorem 13.4 (page 474 of Apostol)we know that given two points there is a unique line between them which is the set of points

So, the set of points on are all points of the form for .

1. The point is not on since there is no such that .
2. The point is on since if we take then .
3. The point is not on since for any .
4. The point is on since if we take then .
5. The point is on since if we take then .