Hi
$\displaystyle \overrightarrow{AP}^2 = \frac12\:\overrightarrow{AB}\cdot\frac12\:\overrig htarrow{AB}\cdot = \frac14\:\overrightarrow{AB}\cdot\overrightarrow{A B}$
$\displaystyle \overrightarrow{AB} = \overrightarrow{AC} + \overrightarrow{CB} = -\vec{a} + \vec{b}$
$\displaystyle \overrightarrow{AP}^2 = \frac14\:\left(-\vec{a} + \vec{b})^2\right) = \frac14\:\left(\vec{a}^2 + \vec{b}^2 -2 \vec{a} \cdot \vec{b}\right)$
$\displaystyle \overrightarrow{CP} = \frac12\:\left(\overrightarrow{CA} + \overrightarrow{CB}\right) = \frac12\:\left(\vec{a} + \vec{b}\right)$
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AB coordinates are (-3,3,-4)
Therefore one parametric equation of l is :
x = -3t + 1
y = 3t -2
z = -4t +3
C(x,y,z) is on the plane iff 2x+y-3z=9
C is on l iff there exists t such that
x = -3t + 1
y = 3t -2
z = -4t +3
Substitute x,y,z in the Cartesian equation of the plane to get one linear equation. Solve for t and substitute in x,y,z expressions.
