# Questions related to vectors and linear algebra

1) Given vectors $u = (1, 1, -2)$ and $v = (0, 1, 1)$, find the value of $t$ such that magnitude of $u + t(v)$ has the smallest value

I have no idea how to begin.

2) Given vectors $u = (1, 1, -2)$, $v = (0, 1, 1)$ and $w = (t, 2, -1)$ where $t$ is any real number. For what values of $t$ are there no scalars $x1, x2$ and $x3$ such that

$x1u + x2v + x3w = e_3$ where vector $e_3 = (0,0,1)$

I've set up

$[1 0 t;1 1 2;-2 1 -1]*[x1;x2;x3] = [0;0;1]$

and I get the sense that I'm supposed to reduce or solve the linear system to get a value for $t$ where there is no solution but am unable to proceed.

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Hint to begin : $||u + tv||^2 = ||u||^2 + 2t<u,v> + t^2||v||^2$ which is a polynomial in $t$.

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How would I obtain 2t(u)(v)? I do not know how to perform such an operation. – Kev Feb 15 '13 at 22:27
I also still do not see how this would help me. – Kev Feb 15 '13 at 22:28