1. **Problem statement:** We have a line $L$ from part 2 (assumed parametric form $x=1+t$, $y=2-t$, $z=3+2t$) and a plane $P$ with equation $x + 3y - z = -4$. (a) Show $L$ intersects $P$ at a unique point and find its coordinates. (b) Find line $M$ in plane $P$, perpendicular to $L$, passing through the intersection point. 2. **Step (a): Find intersection of $L$ and $P$** - Parametric equations of $L$: $x=1+t$, $y=2 - t$, $z=3 + 2t$. - Plane $P$: $x + 3y - z = -4$. Substitute $L$ into $P$: $$ (1+t) + 3(2 - t) - (3 + 2t) = -4 $$ Simplify: $$ 1 + t + 6 - 3t - 3 - 2t = -4 $$ $$ (1 + 6 - 3) + (t - 3t - 2t) = -4 $$ $$ 4 - 4t = -4 $$ Solve for $t$: $$ -4t = -8 \Rightarrow t = 2 $$ Find coordinates at $t=2$: $$ x=1+2=3, \quad y=2-2=0, \quad z=3+4=7 $$ So, $L$ intersects $P$ at unique point $(3,0,7)$. 3. **Step (b): Find line $M$ in $P$, perpendicular to $L$, through $(3,0,7)$** - Direction vector of $L$ is $oldsymbol{d_L} = (1, -1, 2)$. - Plane $P$ normal vector is $oldsymbol{n_P} = (1, 3, -1)$. Since $M$ lies in $P$, its direction vector $oldsymbol{d_M}$ must be perpendicular to $oldsymbol{n_P}$: $$ \boldsymbol{d_M} \cdot \boldsymbol{n_P} = 0 $$ Since $M$ is perpendicular to $L$: $$ \boldsymbol{d_M} \cdot \boldsymbol{d_L} = 0 $$ Find $oldsymbol{d_M}$ satisfying both: Solve system: $$ \begin{cases} d_{Mx} + 3 d_{My} - d_{Mz} = 0 \\ d_{Mx} - d_{My} + 2 d_{Mz} = 0 \end{cases} $$ From first: $$ d_{Mx} = -3 d_{My} + d_{Mz} $$ Substitute into second: $$ (-3 d_{My} + d_{Mz}) - d_{My} + 2 d_{Mz} = 0 $$ $$ -4 d_{My} + 3 d_{Mz} = 0 \Rightarrow 4 d_{My} = 3 d_{Mz} \Rightarrow d_{My} = \frac{3}{4} d_{Mz} $$ Substitute back: $$ d_{Mx} = -3 \times \frac{3}{4} d_{Mz} + d_{Mz} = -\frac{9}{4} d_{Mz} + d_{Mz} = -\frac{5}{4} d_{Mz} $$ Choose $d_{Mz} = 4$ for simplicity: $$ d_{My} = 3, \quad d_{Mx} = -5 $$ Direction vector of $M$ is $\boldsymbol{d_M} = (-5, 3, 4)$. Parametric equations of $M$ through $(3,0,7)$: $$ x = 3 - 5s, \quad y = 0 + 3s, \quad z = 7 + 4s $$ **Final answers:** (a) Intersection point: $(3,0,7)$. (b) Line $M$ parametric form: $$ x=3 - 5s, \quad y=3s, \quad z=7 + 4s $$