Discussion questions · Section 1 of 2

Find the vector equations of the plane $\Pi$ in parametric form and scalar product form, and also the Cartesian equation of the plane $\Pi$ if

(a) $\Pi$ passes through the three points $A(1,0,2)$, $B(0,1,3)$, $C(2,-1,4)$.

(b) $\Pi$ contains the two lines $l_1:\mathbf r=(1+\lambda)\mathbf i+(1+2\lambda)\mathbf j-\lambda\mathbf k$, $\lambda\in\mathbb R$, and $l_2:\mathbf r=(2-\mu)\mathbf i+\mu\mathbf j+(2-2\mu)\mathbf k$, $\mu\in\mathbb R$.

(c) $\Pi$ contains the point $D(1,-1,2)$ and the line $l:\mathbf r=(1+3\lambda)\mathbf i-(1+2\lambda)\mathbf j+6\lambda\mathbf k$, $\lambda\in\mathbb R$.

(d) $\Pi$ contains the point $A(1,0,2)$ and parallel to the y-z plane.

Determine whether each line intersects the plane at a unique point, lies on it, or does not intersect it. If there is a unique intersection, find that point and the angle between the line and plane. All parameters are real.

(a) $\Pi:\mathbf r\cdot(2,1,-3)=5$, $l:\mathbf r=(3,-1,1)+\lambda(-2,1,-3)$.

(b) $\Pi:\mathbf r\cdot(2,1,-3)=5$, $l:\mathbf r=(2,1,0)+\lambda(1,1,1)$.

The line $l$ passes through $A(-1,2,3)$ and $B(5,14,11)$. The plane $p$ has equation $2x+3y-6z=-7$.

(i) Show $l$ is parallel to, but not contained in, $p$.

(ii) Find the distance of $l$ from $p$.

(iii) Find a Cartesian equation of the plane containing $l$ and perpendicular to $p$.

A plane passes through $(1,-2,1)$ and $(2,-1,-1)$ and is parallel to the line $\mathbf r=(1,-1,0)+s(3,1,-2)$, $s\in\mathbb R$.

(i) Find the foot of the perpendicular from $A(1,1,-1)$ to the plane.

(ii) Hence or otherwise find the shortest distance from A to the plane.

(iii) Find the image of A reflected in the plane.

Given $l:\mathbf r=(10,4,2)+\lambda(9,-1,0)$, $\lambda\in\mathbb R$, and $\Pi:\mathbf r\cdot(3,-1,2)=2$, find

(i) the point of intersection of l and $\Pi$;

(ii) a vector equation of the reflection of l in $\Pi$.

For each pair of planes, find their line of intersection and acute angle. All parameters are real.

(a) $\mathbf r\cdot(1,-1,3)=3$ and $\mathbf r\cdot(2,-1,2)=5$.

(b) $\mathbf r=(1,2,7)+\lambda(1,1,0)+\mu(1,0,1)$ and $\mathbf r=(1,2,7)+s(1,2,4)+t(0,1,1)$.