Specialist Mathematics · Unit 3 · Vectors in two and three dimensions · Vectors in three dimensions
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A rectangular storage container has one corner at the origin and edges parallel to the coordinate axes. The opposite corner (diagonally opposite) is located at point C(8, 6, 4). Point P lies on the edge from the origin to vertex C such that P is one-quarter of the way along this edge. Calculate the coordinates of point P.
A rectangular storage container has its base in the xy-plane with one corner at the origin. The container extends 8 units along the positive x-axis, 6 units along the positive y-axis, and 5 units along the positive z-axis. a) State the coordinates of the vertex of the container that lies on the positive z-axis. (1 mark) b) The point P is located at the centre of the top face of the container. Determine the coordinates of P. (1 mark) c) A point Q lies on the interior diagonal of the container that runs from the origin to the opposite corner. Determine the coordinates of the point that is one-third of the way along this diagonal from the origin. (1 mark)
A spacecraft tracking system records the positions of three satellites at a particular time. The table below shows the coordinates \((x, y, z)\) in thousands of kilometres relative to a ground station, where \(x\) represents east-west displacement, \(y\) represents north-south displacement, and \(z\) represents altitude. (a) Determine the distance between Satellite A and Satellite B, correct to one decimal place. (2 marks) (b) Show that the midpoint M of the line segment joining Satellite A and Satellite C has coordinates \((4, -1, 6)\). (2 marks) (c) Hence determine the distance from M to Satellite B, correct to one decimal place. (1 mark)
The point \(P\) lies on the line segment joining \(A(3, -2, 7)\) and \(B(-5, 4, 1)\) such that \(AP:PB = 3:1\). (a) Determine the coordinates of \(P\). (b) Calculate the exact distance from the origin \(O\) to point \(P\). (c) Hence, determine the coordinates of the point \(Q\) on the line segment \(OP\) that is exactly \(5\) units from the origin.