lines in space

**cowboys111** · September 18th 2008, 10:03 AM

Hi, Im trying to find a parametric equation of a line through (2,3,0) and perpendicular to vectors u = i+2j+3k and v = 3i+4j+5k?
To do this I think i need a parallel vector, how do i get this? is it something to do with the cross product?
thanks.

**Laurent** · September 18th 2008, 01:06 PM

Both of your ideas are good, just go ahead: $\vec{u}\times\vec{v}$ is a vector orthogonal to both $\vec{u}$ and $\vec{v}$ . Since the orthogonal complement of a plane in $\mathbb{R}^3$ is a line, $\vec{u}\times\vec{v}$ is a directing vector for the line you want. So a parametric equation for this line is: $\lambda\mapsto (2,3,0)+\lambda \vec{u}\times\vec{v}$ , defined for $\lambda\in\mathbb{R}$ .

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