# Finding Equation of Plane given Vector of Point and Line

See part (ii)

Can I say in formula

$$r=a+\lambda u + \mu v$$

$u \text{ and } v$ are any vectors? Because I was thinking along the lines that $r = a + \lambda AC + \mu AB$ meaning if I have AB, I still need to calculate AC? In the answer, the direction vector is used as it is. So do I conclude that u and v are any vectors?

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No, $\bf u$ and $\bf v$ must be vectors that are in the plane. The plane contains the line. So, the plane contains the direction vector of the line. – David Mitra Nov 13 '11 at 9:56

For the plane with equation $${\bf r}={\bf a}+\lambda{\bf u}+\mu{\bf v}$$ $\bf u$ and $\bf v$ must be non-parallel vectors that are in the plane (or, more precisely, parallel to the plane). Any two non-parallel vectors in the plane will do. You need to find two such vectors first to find the equation of the plane.
For (ii), the plane contains the line; so, the plane contains the direction vector of the line. This is ${\bf v}$ in (ii).
The other vector needed was found from the point $A$ and from a point on the line (point $B$ in (ii)).