Show that the vectors (-1,2,3), (4,1,-2) and (-14,-1,16) do not lie on the same plane

But when you connect the origin to each of the points, then they become vectors. Plus that's exactly how the question is written in the textbook.

To show that the vectors (-1,2,3), (4,1,-2), and (-14,-1,16) do not lie on the same plane, we can use a couple of different methods. One approach is to check if the vectors are linearly dependent. If the vectors are linearly dependent, then they would lie on the same plane. If they are linearly independent, then they do not lie on the same plane.

To check for linear dependence, we can create a matrix using the given vectors as the columns:

[-1 4 -14]
[ 2 1 -1]
[ 3 -2 16]

We can now perform row reduction to check for linear dependence:

Step 1: Subtract 2 times the first row from the second row, and add 3 times the first row to the third row.
[1 -2 6]
[0 9 -32]
[0 1 -5]

Step 2: Divide the second row by 9.
[1 -2 6]
[0 1 -32/9]
[0 1 -5]

Step 3: Subtract the second row from the third row.
[1 -2 6]
[0 1 -32/9]
[0 0 43/9]

Step 4: Divide the third row by 43/9.
[1 -2 6]
[0 1 -32/9]
[0 0 1]

Since the matrix is in row-echelon form and has three non-zero rows, it indicates that the given vectors are linearly independent. Hence, the vectors (-1,2,3), (4,1,-2), and (-14,-1,16) do not lie on the same plane.

No offense but those aren't vectors, those are coordinates.