what is the projection of point in 3d plane

Question

If the coordinate $T(3,4,3) ,B(6,0,0),C(6,0,6)$ and $P(2,0,0)$ in $3$ -dimensions. What is the coordinate of the foot altitude of $P$ with respect to plane $TBC$.

My-Progress:

I am trying to use the equation of plane before resulting $4x+3y=24$. What I need to do next or this step is unnecessary.

Answer 1

You are correct that the plane containing $T$, $B$, and $C$ has equation $4x+3y=24$. So a vector perpendicular to that plane is $(4,3,0)$. Multiplying that vector by some real number $k$ will give the vector from $P$ to its projection $Q = (x_Q,y_Q,z_Q)$ on the plane:

$$ k(4,3,0) = (x_Q,y_Q,z_Q) - (2,0,0) $$ $$ \begin{align*} x_Q &= 4k+2 \\ y_Q &= 3k \\ z_Q &= 0 \end{align*} $$

Since point $Q$ is in the plane, its coordinates satisfy the plane's equation:

$$ 4x_Q + 3y_Q=24 $$ $$ 4(4k+2) + 3(3k) = 24 $$ $$ k = \frac{16}{25} $$ $$ Q = \left(\frac{114}{25}, \frac{48}{25}, 0\right) $$

Answer 2

Write the vector parametric equation of the plane. This is

$r(t,s) = T + t (B - T) + s (C - T) $

Substituting

$r(t,s) = (3,4,3) + t (3, -4, -3) + s(3, -4, 3) $

So the two vectros spanning the plane are $\mathbf{v_1} = [3, -4, -3]^T $ and $\mathbf{v_2} = [3, -4, 3]^T$.

Define the $3 \times 2 $ matrix $A$ as follows

$ A = [\mathbf{v_1, v_2} ] = \begin{bmatrix} 3 && 3 \\ -4 && -4 \\ -3 && 3 \end{bmatrix} $

Then the projector onto the plane is

$P = A (A^T A)^{-1} A^T $

Such the projection of a point $\mathbf{p}$ onto the plane is given by

$ \text{Proj}(\mathbf{p}) = T + P (\mathbf{p} - T) $

Let's compute the projection matrix $P$. We have

$ (A^T A)^{-1} = \dfrac{1}{900} \begin{bmatrix} 34 && - 16 \\ -16 &&34 \end{bmatrix} $

Therefore,

$P = \begin{bmatrix} 0.36 && -0.48 && 0 \\ -0.48 && 0.64 && 0 \\ 0 && 0 && 1 \end{bmatrix} $

And finally, the projection of $Q (2, 0, 0)$ is

$ \text{Proj}(Q) = (3,4,3) + P (Q - (3,4,3) ) = (3,4,3) + P (-1, -4, -3) \\= (3,4,3) + ( 1.56, -2.08, -3 ) = (4.56, 1.92, 0) $

Answer 3

Eq of PN is $(x-2)/3=y/4=z/0=t$, them foot N is $(3t+2,4t,0)$ which will lie on the plane so $9t+6+16t=24\implies t=18/25$. So the foot $N=(104/25,72/25,0)$.