समीकरण r = (i + 2j + k) + lambda*(i - j + k) तथा r = (2i - j - k) + mu*(2i + j + 2k) वाली रेखाओं के बीच की न्यूनतम दूरी ज्ञात कीजिए।
Get the complete, step-by-step math solution for: "समीकरण r = (i + 2j + k) + lambda*(i - j + k) तथा r = (2i - j - k) + mu*(2i + j + 2k) वाली रेखाओं के बीच की न्यूनतम दूरी ज्ञात कीजिए।...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify the given vectors
The given equations of the lines are in the form \mathbf{r} = \mathbf{a} + $\lambda$\mathbf{b}$. We identify the position vectors \mathbf{a_1} and \mathbf{a_2} of points on the lines, and the direction vectors \mathbf{b_1} and \mathbf{b_2} of the lines.
Step 2: Calculate $(\mathbf{a_2} - \mathbf{a_1})$
We calculate the vector connecting a point on the first line to a point on the second line by subtracting \mathbf{a_1} from \mathbf{a_2}$. This vector is crucial for finding the shortest distance.
Step 3: Calculate $(\mathbf{b_1} \times \mathbf{b_2})$
To find the shortest distance between two skew lines, we need a vector perpendicular to both direction vectors. This is achieved by calculating the cross product of \mathbf{b_1} and \mathbf{b_2}$.
Step 4: Calculate $|\mathbf{b_1} \times \mathbf{b_2}|$
We find the magnitude of the cross product vector. This magnitude will be used in the shortest distance formula.
Step 5: Calculate $(\mathbf{a_2} - \mathbf{a_1}) \cdot (\mathbf{b_1} \times \mathbf{b_2})$
We compute the scalar triple product of the vector connecting the points and the cross product of the direction vectors. This dot product gives us the numerator for the shortest distance formula.
Step 6: Apply the shortest distance formula
Finally, we use the formula for the shortest distance between two skew lines, which is the absolute value of the scalar triple product divided by the magnitude of the cross product. We rationalize the denominator for the final answer.