Home » Blog » Find all values of t so that (1,t,1), (t,1,0), (0,1,t) are dependent

Find all values of t so that (1,t,1), (t,1,0), (0,1,t) are dependent

Find all t \in \mathbb{R} such that the vectors

    \[ (1,t,1), \qquad (t,1,0), \qquad (0,1,t) \]

are linearly dependent.


The vectors (1,t,1), \ (t,1,0), \ (0,1,t) are linearly dependent if and only if the scalar triple product

    \[ (1,t,1) \cdot (t,1,0) \times (0,1,t) = 0. \]

So, we compute

    \begin{align*}  (1,t,1) \cdot (t,1,0) \times (0,1,t) &= (1,t,1) \cdot (t - 0, 0 - t^2, t - 0) \\  &= (1,t,1) \cdot (t,-t^2,t) \\  &= (1)(t) + (t)(-t^2) + (1)(t) \\  &= 2t-t^3. \end{align*}

So, the vectors are dependent for all t such that 2t-t^3 = 0. Thus the vectors are dependent for

    \[ t = 0, \ \pm \sqrt{2}. \]

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):