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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
up vote 0 down vote accepted

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)).

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Nope. The vectors "u" and "v" must be parallel to the said plane and must not be parallel to each other. In this case, since Vectors "a","b" and "c" are given, the only way to determine vectors parallel to the plane(or in this case contained in the plane), Would be to calculate vectors AB AC or BC. Any two would suffice.

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