Question

Find the angle between each of the following pairs of lines:
1, 1, 2; $square root of 3 minus 1 comma space space minus square root of 3 minus 1 comma space 4$

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Solution

The direction-ratios of the two lines are 1,1,2 and

square root of 3 minus 1 comma space space minus square root of 3 minus 1 comma space 4

Let θ be the angle between the lines.

therefore space space space cos space straight theta space equals space fraction numerator left parenthesis 1 right parenthesis thin space left parenthesis square root of 3 minus 1 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis negative square root of 3 minus 1 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis 4 right parenthesis over denominator square root of 1 plus 1 plus 4 end root square root of 3 plus 1 minus 2 square root of 3 plus 3 plus 1 plus 2 square root of 3 plus 16 end root end fraction space space space space space space space space space space space space space space space space space space space equals fraction numerator square root of 3 minus 1 minus square root of 3 minus 1 plus 8 over denominator square root of 6 space square root of 24 end fraction space equals space fraction numerator 6 over denominator square root of 6. space 2 square root of 6 end fraction space therefore space space space cos space straight theta space equals space 1 half space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space straight theta space equals space 60 degree space space space space space space space space space space space space

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