Practice section 3 of 7

Find the line of intersection of

$$\Pi_1:\quad\mathbf r=(1,2,3)+\alpha(1,2,1)+\beta(2,2,1)$$

and

$$\Pi_2:\quad\mathbf r=(1,2,3)+s(1,1,0)+t(4,6,7),$$

where all four parameters are real.

Find the acute angle between the planes

$$\begin{aligned}\Pi_1:\quad\mathbf r\cdot(\mathbf i-\mathbf j-\mathbf k)&=6,\\\Pi_2:\quad\mathbf r\cdot(\mathbf i-\mathbf j-2\mathbf k)&=10.\end{aligned}$$

A computer-controlled machine makes plane cuts by using the equation of the cutting plane, and drills straight holes by using the equation of a drill line.

A $10\text{ cm}\times20\text{ cm}\times30\text{ cm}$ cuboid is positioned relative to the x-, y- and z-axes as in Figure 1. The diagram fixes the positive directions: $OA=10$ cm along positive y, $AB=20$ cm along negative x, and $BF=30$ cm along positive z.

First a plane cut removes the corner at E. It passes through P, Q and R, the midpoints of ED, EA and EF respectively.

(i) Show that $\overrightarrow{PQ}=(0,5,-15)$ and $\overrightarrow{PR}=(-10,5,0)$.

(ii) Find the Cartesian equation of the plane p containing P, Q and R.

(iii) Find the acute angle between p and the plane DEFG.

A hole is then drilled perpendicular to triangle PQR, as in Figure 2. It passes through T, which divides PS in the ratio $PT:TS=4:1$, where S is the midpoint of QR.

(iv) Show that T has coordinates $(-4,9,24)$.

(v) State the vector equation of the drill line.

(vi) If drilling continues through the cuboid along that line, determine whether the drill exits from plane OABC or ODGC. Justify your answer.

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 $p$.