2021 RI Promo Q9

Distances in this question are in metres.

Harry and Tom’s model airplanes are taking off from the horizontal ground, which is the \(x-y\) plane. Tom’s airplane takes off after Harry’s. The position of Harry’s airplane \(t\) seconds after it takes off is given by \(\mathbf{r}=\left( 5\mathbf{i}+6\mathbf{j} \right)+t\left( -4\mathbf{i}+2\mathbf{j}+4\mathbf{k} \right)\). The position of Tom’s airplane \(s\) seconds after it takes off is given by \(\mathbf{r}=\left( -39\mathbf{i}+44\mathbf{j} \right)+s\left( 4\mathbf{i}-6\mathbf{j}+7\mathbf{k} \right)\).

Extension question:

Interpret the geometrical meaning of the vector \(-4\mathbf{i}+2\mathbf{j}+4\mathbf{k}\) in the equation of the position of Harry's airplane.

1. State the height of Harry’s airplane two seconds after it takes off and find its distance travelled in the two seconds.[3]
2. Find the acute angle between the path of Harry’s airplane and the ground.[2]
3. Show that the paths of the airplanes are perpendicular.[1]
4. Given that the two airplanes collide, find the coordinates of the point of collision. How long after Harry’s airplane takes off does Tom’s airplane take off?[3]
5. Find the cartesian equation of the plane in which both paths of the airplanes lie.[3]