Need help to find out what im doing wrong. Below is the question and my attempts. Given tree points: A = (1,3) B = (2,-6) C = (3, t)
a) Find t so that vector AB is perpendicular to vector BC
vector AB = [2, -18] Vector BC = [6, -6t] 12 + 108t = 0 t = -9
b) Find t so that vector |AC| = sqrt(13) vector AC = [3,3t]
sqrt(3^2 + 3^2*t^2) = sqrt(13) t = sqrt(22/9)
c) Find t so that [3, -4 ] = k * vector AB + vector BC
stuck on this
The first problem is that the vectors you're using are different from $\vec{AB}$ and $\vec{BC}$ respectively. Given the points $(a,b)$ and $(c,d)$ in the plane, then the vector carrying us from the first point to the second point is given by the differences in the coordinates of the points, starting with the coordinates of the endpoint. Thus, our vector here is $(c-a,d-b).$ Applying this tells us that $\vec{AB}=(1,-9)$ and $\vec{BC}=(1,t+6).$ If these are perpendicular, then their dot product vanishes, so that we have $$(1,-9)\cdot (1,t+6)=0.$$ Can you complete this now?