HI I want to check if my answers are right here, I can't do the last part so I believe my answers in the previous parts must be wrong..

Points A,B,C,D have the following coordinates

A:(1,3,1) B:(1,2,4) C:(2,3,6) D:(5,-2,1)

ai.) Evaluate the vector product, giving your answers in terms of the unit vectors i,j,k.

I found AB and AC and got -5i, 3j and 1k from doing the cross product.

aii.) Find the area of the triangle ABC.

I took the magnitude of the cross product result in ai.) and halved it, getting (1/2)sqrt(35)units^2

b.) The plane containing the points A,B, C is denoted by pi and the line passing through D perpendicular to pi is denoted by L. The point of intersection of L and pi is denoted by P.

bi.) FInd the cartesian equation of pi.

AB is the direction vector of pi, I thought. And then I'll need a point so

I got x=1, y=3-t, z=1+3t

bii.) FInd the cartesian equation of L.

The result of the cross product in ai.) is the direction vector of L, I thought. Dotting it by the direction vector of pi, I will get a zero.

So my answer was

x=5-5t, y=-2+3t, z=1+t

c.) Find the coordinates of P.

This is where I know my previous answers are wrong. I though to equate all the x, y and z values of L and pi. I get different values for t each time...

Please help me out!

Thank you!