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 .

- ;
- ;
- ;
- ;
- .

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

- 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.

- 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.

- 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.

- 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.

- 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.