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# 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.