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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. – Bill Cook Feb 8 '12 at 15:23

I'm not exactly certain I've understood your question correctly, but here is an attempt; I'll interpret the problem as follows: let $U = (U_1, U_2, U_3)$ and $V = (V_1, V_2, V_3)$ be vectors in $\mathbb{R}^3$. Find a vector $R = (R_1, R_2, R_3)$ that is perpendicular to both $U$ and $V$. This is the same as requiring that $U\cdot R = V \cdot R = 0$, or equivalently, that the vector $R$ solves the following linear system: \begin{align*} U_1 R_1 + U_2 R_2 + U_3 R_3 &= 0 \\ V_1 R_1 + V_2 R_2 + V_3 R_3 & = 0 \end{align*} Plug in the vectors $U$ and $V$ and solve using elimination for the unknowns $R_1$, $R_2$ and $R_3$. The system has more unknowns than equations, and hence it always has a nontrivial solution (in fact, infinitely many). If the vector $U$ is not a scalar multiple of $V$ (in other words, if the vectors are linearly independent), the solution space is $1$-dimensional, consisting of all scalar multiples of the cross product.

If, however, you require the actual cross product, rather than just a vector perpendicular to the ones given, there are additional complications (you need to determine the length and direction), and my guess is that it would be easier to just compute the cross product directly.

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