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If I have 2 lines with its symmetric equations I can get the vectors U and V of each line, and with a cross product I can get the vector R; but how can I get the vector R without a cross product?

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    What do you mean "the vector R"? What is R? If you mean, "How can a find a vector, R, which is perpendicular to both U and V?" Then the cross product is the easiest way to do so. Without using the cross product, you'll end up having to solve a linear system of equations.2012-02-08

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