Submitted by arobx on Sat, 12/23/2017 - 18:49
Forums:
I have two questions about the first part of the question ( and the solution given on the 'Vectors' S2 module):
1. When expressing $m_{3/4}$ as a general 3D vector, why can we just assume that the direction is the same for both vectors wouldn't we need $a\vec{i}+b\vec{j}+c\vec{k}$ for $m_3$ say and then $k\vec{i}+m\vec{j}+n\vec{k}$ for $m_4$? This is why I got stuck initially.
2. Furthermore another reason I couldn't solve this problem was the use of 'WLOG a=1', why can we just do this i.e. why is there no loss in generality, does the direction not depend on the value of a? How will I know in future problems if I can use WLOG?
Sorry for the delay!
We start by assuming that the directions are of the form $a{\bf i} + b {\bf j} + c {\bf k}$, and hope that we will find 2 different solutions. Using the dot product with both $m_1$ and $m_2$ results in the equations $2ab=c^2$ and $2ac=b^2$. $a$ cannot be equal to zero (as then both $b$ and $c$ are equal to zero), and since we are looking for ${\bf directions}$ we can scale so that there is a component of $1$ in the ${\bf i}$ direction.
(Note that ${\bf i} + {\bf k}$, $5{\bf i} + 5{\bf k}$, $-{\bf i} - {\bf k}$ and $217{\bf i} + 217{\bf k}$ are all in the same direction. You can keep everything in terms of $a$ if you prefer. This would give the answers $a{\bf i}$ and $a{\bf i}+2a{\bf j}+2a {\bf k}$.)
Hope this helps, let me know if you need any more.
Happy New Year!