I am working on Chapter 9, Example 24, from Pearson Core Pure Mathematics book 1.
'Find the shortest distance between the parallel lines with equations 1: $r=i + 2j - k + \lambda(5i + 4j + 3k)$ and 2: $r=2i+k+\mu(5i+4j+3k)$, where $\lambda$ and $\mu$ are scalars'.
The answer starts by finding the difference between two generalised points on A and B lines 1 and 2, with the expression:
$AB = (i - 2j + 2k) + t(5i+4j+3k)$ where $t = \mu - \lambda$.
I understand subtraction of the starting point vectors to get a point vector on the subtrated line, i.e. $(2i+k) - (i + 2j - k) = (i - 2j + 2k)$. But I cannot understand how the $t = \mu - \lambda$ operation is allowed, either intuitively or mathematically.
e.g. mathematically, subtracting both line 1 and line 2:
$(i - 2j + 2k) + - \mu(5i+4j+3k) - \lambda(5i + 4j + 3k)$. I dont see here how the $\mu$ and $\lambda$ can factor out to the t expression
I missed it, just factorising:
$\mu(5i + 4j + 3k) - \lambda(5i + 4j + 3k)$
$5i\mu + 4j\mu + 3k\mu - 5i\lambda -4j\lambda - 3k\lambda$
$(5i\mu - 5i\lambda) + (4j\mu -4j\lambda) + (3k\mu-3k\lambda)$
$5i(\mu - \lambda) + 4j(\mu-\lambda) + 3k(\mu - \lambda)$
$(\mu - \lambda)(5i + 4j + 3k)$