+108 Abstracting from a physics problem here...
In short, need to show that a plane (defined by specific indices) is perpendicular to a vector.
The vector is:
\( \overline{m} =h \widehat{a} +k \widehat{b} +l \widehat{c}\)
And the plane can be spanned by:
\( \overline{v_1} =\frac{a}{h} \widehat{a} -\frac{a}{k} \widehat{b} \)
\( \overline{v_2} =\frac{a}{h} \widehat{a} -\frac{a}{l} \widehat{c} \)
Now, for h,k or l not equal to 0, this works (taking the dot product of either vector basis vector with vector m gives you 0). However how can we prove this for h,k or l = 0 (taking note that only 1 of these terms can equal 0, as if 2 equal zero we cannot define a plane).
This seems to work, but am I committing a math crime here?
\( \lim_{l \rightarrow 0} [ \overline{v_2} . \overline{m} ]= \lim_{l \rightarrow 0} [ \frac{a}{h}h-\frac{a}{l}l ]=a-a=0\)
