Virtual Reality -- Vectors Spaces, I -- Answers

Yes. We must solve the equation
```
  t (-1, -1) + (1 - t) (5, 7) = (2, 3)
(-t, -t) + (5 - 5 t, 7 - 7 t) = (2, 3)
           (5 - 6 t, 7 - 8 t) = (2, 3)
```
which leads to the pair of equations
```
5 - 6 t = 2
7 - 8 t = 3
```
Working with the first equation we see that
```
5 - 6 t = 2
   -6 t = -3
      t = 1/2
```
and with the second equation
```
7 - 8 t = 3
   -8 t = -4
      t = 1/2.  
```
Since these two equations have the same solution t = 1/2 the point (2, 3) is on the line determined by the two points (-1, -1) and (5, 7). Furthermore since t = 1/2 it is exactly halfway between these two points.
No. We must solve the equation
```
  t (-1, -1) + (1 - t) (5, 7) = (3, 4)
(-t, -t) + (5 - 5 t, 7 - 7 t) = (3, 4)
           (5 - 6 t, 7 - 8 t) = (3, 4)
```
which leads to the pair of equations
```
5 - 6 t = 3
7 - 8 t = 4
```
Working with the first equation we see that
```
5 - 6 t = 3
   -6 t = -2
      t = 1/3
```
and with the second equation
```
7 - 8 t = 4
   -8 t = -3
      t = 3/8.  
```
Since these two equations have different solutions the point (3, 4) is not on the line determined by the two points (-1, -1) and (5, 7).
No.
Yes.
u + t v