# 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.