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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

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Solution

The equations of the given lines are

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

Comparing the given equations, we obtain

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

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

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Q. Find the shortest distance between the lines.

\to r =^i + 2^j +^k + λ (^i -^j +^k)

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

Q. Find the shortest distance between the lines

\to r =^i +^j + λ (2^i -^j +^k)

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

Q. Find the shortest distance between the following pairs of parallel lines.

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

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

Q. Find the shortest distance between line

l_{1}

l_{2}

whose vector equations are

\to r =^i +^j + μ (2^i -^j +^k)

................ (i)
and

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

............. (ii)

Find the angle between the following pairs of lines:

(i)

(ii) and

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

Standard XII Mathematics

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