Question

If `bar u`, `bar v`, and `bar w` are linearly independent vectors, find the values of `t`?

`(t^2-t-2)baru+(2t^2-3t-2)bar v+(3t^2-5t-2)bar w=0`

Answer 1

`t = 2`

Explanation:

If these vectors are linearly independent, then:

`alpha bb u + beta bb v + gamma bb w = bb 0 implies alpha, beta, gamma = 0`

Or in this case:

• `alpha(t), beta(t), gamma(t) = 0`

Factoring:

• `alpha(t) = t^2-t-2 = (t-2)(t+1)`

• `beta(t) = 2t^2-3t-2= (t - 2) (2 t + 1)`

• `gamma(t) = 3t^2-5t-2 = (t - 2) (3 t + 1)`

`implies t = 2`