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.