I need to find the equation of the plane that contains point $(1,0,0)$ and line $r=(1+\lambda)i + 3j + 2\lambda k$
Correct Answer
Plane parallel to $(1,3,0)-(1,0,0)=(0,3,0)$
Plane perpendicular to $(1,0,2)\times (0,3,0) = (6,0,3)$
Then
$r \cdot (2,0,1) = 2$
Try 1
So I first thought to find the normal
$n = (1,0,0) \times (1,0,-2) = (0,2,0)$
Then I have
$r\cdot (0,2,0) = (1,0,0) \cdot (0,2,0)$
$r\cdot (0,2,0) = 0$
Which is wrong?
Try 2
I also tried using the formula ${\bf r}={\bf a}+\lambda{\bf u}+\mu{\bf v}$
$r = (1,0,0) + \lambda (1,0,0) + \mu (1,0,-2)$
Which is wrong too. Why is that?
How can I express the plane in normal form and ${\bf r}={\bf a}+\lambda{\bf u}+\mu{\bf v}$? How do convert between them? And whats wrong my my tries?