Exam time

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) y = m x + c is useful only in two dimensions.

(2) Defining a line as collection of position vectors given by c \vec{v}, where \vec{v} can be a vector in any dimensions and c \in R. For example c(1,0,0) with c \in R 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 2 x + y - z = 0 and x + y + 2 z= 0 requires that z = x+y, and x = -y substituting these conditions for the coordinates of a general point (x,y,z) gives (x,y,z) = (x, -x, 0) = x (1,-1,0) The equation of a general line (not necessarily passing through the origin) can be written as \vec{a} + c \vec{v} where \vec{a} 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 ax + b y + c z = 0, insist that the given points satisfy the equation and solve for a,b,c.

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 \vec{b} 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 \vec{b} is chosen so that two of there equations are different then the matrix equation has no solution.

Author: strangeset

A nomad at heart, I enjoy observing, analysing, connecting, understanding and dreaming. I am a big fan of science and tech. Forever learning and experimenting.

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