Find the shortest distance between the skew lines: r_1 = (i + j + k) + lambda*(2i + 3j + 4k) and r_2 = (2i + 3j + 5k) + mu*(i + 2j + 3k).
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Step-by-Step Solution
Step 1: Identify position vectors and direction vectors
The given equations of the lines are in the form \vec{r} = \vec{a} + \lambda\vec{b}$. We identify the position vectors \vec{a}_{1} and \vec{a}_{2} of points on the lines, and their respective direction vectors \vec{b}_{1} and \vec{b}_{2}$.
Step 2: Calculate $\vec{a_2} - \vec{a_1}$
We calculate the vector connecting a point on the first line to a point on the second line by subtracting the position vector $\vec{a}$_{1} from $\vec{a}$_{2}$. This vector is crucial for finding the shortest distance.
Step 3: Calculate $\vec{b_1} \times \vec{b_2}$
To find the shortest distance between skew lines, we need a vector perpendicular to both direction vectors. This is achieved by calculating the cross product of $\vec{b}$_{1} and $\vec{b}$_{2}$.
Step 4: Calculate $|\vec{b_1} \times \vec{b_2}|$
We find the magnitude of the cross product vector $\vec{b}$_{1} $\times$ $\vec{b}$_{2}$. This magnitude will be used in the shortest distance formula.
Step 5: Apply the shortest distance formula
The shortest distance d between two skew lines is given by the formula d = $\left$| $\frac${($\vec{a}$_{2} - $\vec{a}$_{1}) $\cdot$ ($\vec{b}$_{1} $\times$ $\vec{b}$_{2})}{|$\vec{b}$_{1} $\times$ $\vec{b}$_{2}|} $\right$|$. We substitute the calculated values into this formula to find the distance.