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Conditions for a system of linear equations to have a unique solution

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Question

Find the shortest distance between the lines whose vector equations are

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Solution

The given lines are and

It is known that the shortest distance between the lines, and , is given by,

Comparing the given equations with and , we obtain

Substituting all the values in equation (1), we obtain

Therefore, the shortest distance between the two given lines is units.

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Q. Find the shortest distance between the lines whose vector equations are

\to r = (^i + 2^j + 3^k) + λ (^i - 3^j + 2^k)

\to r = (4^i + 5^j + 6^k) + μ (2^i + 3^j +^k)

Q. The shortest distance between the skew lines

\to r = (^i + 2^j + 3^k) + t (^i + 3^j + 2^k)

\to r = (4^i + 5^j + 6^k) + t (2^i + 3^j +^k)

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Q. Find the shortest distance between the line whose equations are

\to r =^i + 2^j + 3^k + λ (2^i + 3^j + 4^k)

\to r = 2^i + 4^j + 5^k + μ (4^i + 6^j + 8^k)

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l_{1}

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Conditions for a system of linear equations to have a unique solution

Standard XII Mathematics

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