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2. (a) Prove that the lines \( x=-1-t, y=-2+t, z=2+2 t \) and \( x=2+2 t, y=5+3 t, z=5+2 t \) are skew lines, and (b) find the distance between them.

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Final answer:

The provided lines are skew because they are not in the same plane, they are not parallel and they don't intersect. The distance between them can be calculated using a formula that involves the direction ratios and points on the lines.

Step-by-step explanation:

In mathematics, two lines are considered to be skew lines if they are not in the same plane, hence, they are non-coplanar and do not intersect. We can easily determine they are skew by finding if the lines are not parallel nor do they intersect. In this case, you can compare the direction ratios between the two line equations you've given, and see they are not equal, which means the lines aren't parallel. They also don't intersect, so these lines are skew.

To find the distance between two skew lines, we can utilize the formula for the shortest distance, which involves the direction ratios. The formula is:

| c1 - c2 | / sqrt(a^2 + b^2 + c^2)

Where 'c1' and 'c2' are points on the lines, and 'a', 'b', and 'c' are direction ratios of the lines. Plug the given values into the formula to get the distance between lines.

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