Consider the set of numbers 53,54,−51.
(i) State the necessary and sufficient condition for three real numbers to serve as direction cosines of a line in three–dimensional space.
(ii) Using the condition from part (i), test whether the given numbers can represent the direction cosines of a line. Show all steps of your verification.
(iii) If the numbers do form valid direction cosines, identify the octant(s) in which the line can lie, justifying your answer on the basis of the signs of the cosines.