Please help me solve the following question of vectors:

Question

anonymous · Answer

Find 2 values of k such that the following set of equations does not have a unique solution, and solve the equations in the 2 cases.

$$x+y+z = 3$$
$$x+2y+kz=6$$
$$x+ky+(k+2)z = 9$$

anonymous · Answer

so I did (1) - (2) and (1) - (3) to eliminate x, and then solve the equations simultaneously to get k, but I only have one value of k :P

terenzreignz · Answer

This is vectors?

anonymous · Answer

no this isn't shucks

anonymous · Answer

see, my brain isn't working rn.

anonymous · Answer

well it kinda is. 3D planes, you see :P

terenzreignz · Answer

Cheating's easier :D
Here's a nice shortcut
 Let's consider this matrix (of coefficients)
$$\LARGE \left[\begin{matrix} 1&1&1\1&2 &\color{blue}k\1 & \color{blue}k &\color{blue}k+2\end{matrix}\right]$$

anonymous · Answer

right

terenzreignz · Answer

Can you get its determinant?

anonymous · Answer

yup, just give me a few

terenzreignz · Answer

wait what?
There is only one determinant :/

anonymous · Answer

give me a few minutes :P

anonymous · Answer

oh so k = 0, 3

terenzreignz · Answer

huh?

anonymous · Answer

what huh

terenzreignz · Answer

You got its determinant and equated it to zero, then?

anonymous · Answer

yup yup

anonymous · Answer

and then i equate it to each other and do that stuff :P

terenzreignz · Answer

Well then, the rest is history :D

anonymous · Answer

*slow clapping in the distance*

terenzreignz · Answer

By 'history' I mean solving a system of linear equations, which I'd rather not do, because I'm so error prone :/

anonymous · Answer

agreed :)

anonymous · Answer

ok i might be messing this up, but i'm getting no solutions for k = 0, and for k = 3, i'm getting a point or something, with co-ordinates (0,3,0)

anonymous · Answer

and this doesn't seem to be making any sense

terenzreignz · Answer

Well, you're not supposed to get a proper solution in both cases.

anonymous · Answer

ohhhkay, but still, idk something seems wrong.

terenzreignz · Answer

You worry too much... as long as you got that determinant right, there's no reason this should fail... (is there?)

anonymous · Answer

other than the fact that my brain loves to make me make silly mistakes, no, there isn't any reason :P

anonymous · Answer

arrey i just realised: there said no unique solution. it can't be a point. I know what i have to do :P

anonymous · Answer

well, i did another thing, and got no solutions for that too. I give up :P

ganeshie8 · Answer

try, 
k = -2