There seems to be this question that I can't seem to be able to solve. I'm hoping someone could help me figure out how to solve it.
Question: Find the shortest distance between the lines
$\displaystyle\frac {x+1}{7} = \frac {y+1}{-6} = \frac {z+1}{1}$ and $\displaystyle\frac {y-3}{1} = \frac {y-5}{-2} = \frac {z-7}{1}$
Converting it into vector form, you get
$-i -j -k+ \lambda(7i-6j+k)$ and $3i + 5j+ 7k + \upsilon(i -2j + k)$.
Solving it with the formula $\left|\dfrac {(b_1\times b_2)(a_2-a_1)}{|b_1\times b_2|}\right|$ where $a_2$ and $a_1$ represent the two known points and $b_1$, $b_2$ represent the vector parallel to the line. However while solving it i get the answer as $\dfrac {22}{\sqrt{29}}$ but the answer in the book is $2 \sqrt{29}$
I calculated $a_2-a_1$ to be $4i + 6j + 8k$ and $b_1\times b_2$ to be $-4i + 6j - 8k$. The dot product comes out to be $-44$ (Which becomes $44$ since we're taking the modulus). But i can't seem to be able to reduce it further than $\dfrac {22}{\sqrt{29}}$. I'd appreciate any help in figuring out where I went wrong.
$b_1$ x $b_2$ should be $-4i-6j-8k$. So, the dot product is 116 and the result will be $116/\sqrt{116}$, which is $2\sqrt{29}$.