Vectors - collinear and perpendicular

Question

A bird is at point $P$ whose coordinates are $(4, -1, 5)\text{m}$. The bird observes two points $P_1$ and $P_2$ having coordinates $(-1,2,0)$ and $(1,1,4)$ respectively. At time $t = 0$, it starts flying in the plane of three positions with a constant speed of $5 \text{m/s}$ in a direction perpendicular to the straight line $P_1P_2$ till it sees them collinear at time '$t$'. Calculate '$t$'.

For the conditions to be satisfied, the position of the bird at time '$t$' should be collinear to $P_1P_2$ and perpendicular to $P$. However, I don't know where to proceed from there. We are required to solve this question using properties of vectors.

Answer 1

Here are some hints.

• Given three points $a,b,c$ in the plane, how can you calculate two vectors in the plane?
• Given two vectors in the plane, how can you calculate a vector normal $n$ to the plane?
• What conditions must be satisfied for the bird's flight direction $v$ to be both in the plane and perpendicular to the vector defined by the two points the bird is looking at? (It must be perpendicular to two vectors. Which ones?)
• Once the bird reaches the point of colinearity, what conditions are met?
• How is the distance from $P$ to this point calculated? How is the time $t$ calculated from this distance?