2026 HCI Promo Q12

An entertainment company is designing a new indoor laser tag arena. Two players are asked to simulate a match within the arena to test out its design.

Player 1 fires a continuous green laser beam \(l_1\) in the direction \(\begin{pmatrix}2\\2\\1\end{pmatrix}\) from a point \(P(1,2,1)\).

Player 2 fires a continuous red laser beam \(l_2\) in the direction \(\begin{pmatrix}2\\-1\\1\end{pmatrix}\) from a point \(Q(16,8,1)\).

1. Determine whether the two laser beams meet.[2]

A flat reflective board containing the \(3\) points \(A(7,1,3)\), \(B(5,2,4)\) and \(C(6,1,5)\), is placed within the arena.

2. Show that the cartesian equation of the plane that represents the flat reflective board is given by \(2x+3y+z=20\).[3]
3. Find the coordinates of the point \(H\) at which the green laser beam \(l_1\) hits the reflective board.[3]

A target is fixed at \(T\left(5,3,\dfrac{3}{2}\right)\). In order to hit the target \(T\), a player must hit the reflective board in a way such that the reflected laser from the board will hit \(T\). The point \(F\) on the reflective board that is nearest to \(P\) has coordinates \(\left(\dfrac{18}{7},\dfrac{61}{14},\dfrac{25}{14}\right)\).

4. Find the position vector of the point \(P\prime\) which is the reflection of \(P\) in the reflective board.
   Hence, show that the reflected laser beam of \(l_1\) is parallel to the vector \(\begin{pmatrix}8\\19\\4\end{pmatrix}\).[3]
5. Find the equation of the reflected laser beam of \(l_1\). Hence determine if Player 1 managed to hit the target \(T\).[3]