It is the time of exam again and there is the usual stream of students asking for
clarification and solution of different concepts. So here I discuss some common doubts.
Chapter 2: problem 2: Definition of a line passing through the origin in various ways.
(1) is useful only in two dimensions.
(2) Defining a line as collection of position vectors given by , where can be a vector in any dimensions and . For example with would define the x-axis in three dimensions.
(3) Defining a line as intersection of two planes in 3 dimensions, for example the
intersection of the xy-plane and the xz-plane defines the x-axis.
Given any two equations of planes passing through zero, one can find the intersection and express it in terms of scaling of a vector. As an example, the intersection of and requires that , and substituting these conditions for the coordinates of a general point gives The equation of a general line (not necessarily passing through the origin) can be written as where is a constant vector.
Chapter 2: problem 4, how to find a plane that passes through the given three points. Assume a general equation of a plane , insist that the given points satisfy the equation and solve for .
Chapter 3: problem 5, notice that the B matrix has only two rows that are linearly independent. There are 3 variables, hence one variable is free. Thus either there are infinite solutions or there is no solution. When the is chosen such that equations 1 and 3 and equations 2 and 4 are the same, the equation is solvable with infinite solutions, whereas if is chosen so that two of there equations are different then the matrix equation has no solution.