Rory and Duncan want to kick the ball. Both footballs will move in
a straight line path, starting their motion at such times (may be the
same time or different) that they definitely collide. The position of each
football is described with reference to a standard Cartesian coordinate
system, and all distances are measured in centimetres. Football 1 (kicked
by Rory) moves in a straight line path given parametrically by
$x = 6 - 2t, y = 3t, z = 1 + 5t$, where parameter $t \ge 0$.
Football 2 (kicked by Duncan) moves in a straight line path given para-
metrically by
$x = 4 - 2s, y = 2 + 4s, z = 10 + s$, where parameter $s \ge 0$.
1.1 (5) Determine the parameter values $s$ and $t$ where this collision occurs, and
the coordinates of the point of collision.
A photo camera is located at the point with coordinates (30, 30, 50).
1.2 (2) Verify that the camera does not lie on the path of either football 1 or
football 2
Photocamera makes photos the motion of football 2.
1.3 (6) Determine the value of the parameters $s$ and the corresponding coor-
dinates of the point where football 2 would be closest to the camera. Further,
determine how far apart football 2 and the camera would be in this situation.
1.4 (2) With reference to the collision information found in 1.1, and your answers
in 1.3, explain whether or not football 2 will ever reach the point closest to the
camera.
