At 10 a.m. a plane A A\,A has position vector (2i−5j)(2\mathbf{i} - 5\mathbf{j})(2i−5j) km and moves with constant velocity (−4i+6j)(-4\mathbf{i} + 6\mathbf{j})(−4i+6j) km h−1^{-1}−1.
At the same time a plane B B\,B has position vector (−3i−9j)(-3\mathbf{i} - 9\mathbf{j})(−3i−9j) km and moves with constant velocity (i+8j)(\mathbf{i} + 8\mathbf{j})(i+8j) km h−1^{-1}−1.
The unit vectors i\mathbf{i}i and j\mathbf{j}j are due east and due north respectively.
Find the position vector of A A\,A relative to B B\,B at t t\,t hours after 10 a.m.
Find the time at which A A\,A is due west of BBB.
Find the time, after 10 a.m., at which the planes are exactly 37 km apart.