# Finding a result vector from 2 vectors without cross product

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.