3. Determine that the vectors u=[t, 4, 2t+1] and v=[t+2, 1-t, -1] are perpendicular, find the possible values of the contant, t.

t(t+2) + 4(1-t) - (2t+1) = 0

t = 1,3

thank you steve :)

To determine if two vectors are perpendicular, we need to check if their dot product is zero. The dot product of two vectors u and v, denoted as u·v, is given by the formula:

u·v = u₁v₁ + u₂v₂ + u₃v₃

For the given vectors u = [t, 4, 2t+1] and v = [t+2, 1-t, -1], the dot product is:

u·v = (t)(t+2) + (4)(1-t) + (2t+1)(-1)
= t² + 2t + 4 - 4t + 2t + 1 - 2t - 1
= t²

To find the possible values of the constant t, we set the dot product u·v equal to zero:

t² = 0

Solving this equation, we find that t = 0. Therefore, the possible value of the constant t in this case is t = 0.