Discussion questions · Section 2 of 2

The planes $p_1:\mathbf r\cdot(2,-2,1)=1$ and $p_2:\mathbf r\cdot(-6,3,2)=-1$ meet in line l.

(i) Find the acute angle between the planes.

(ii) Find a vector equation for l.

(iii) $A(4,3,c)$ is equidistant from the two planes. Calculate both possible values of c.

The planes $p_1,p_2,p_3$ are $2x-5y+3z=3$, $3x+2y-5z=-5$, $5x+\lambda y+17z=\mu$, respectively. Here $\lambda,\mu$ are constants. When $\lambda=-20.9$, $\mu=16.6$, find x,y,z satisfying all three equations.

The first two planes meet in a line l.

(i) Find a vector equation of l.

(ii) If $p_3$ contains l, find $\lambda,\mu$.

(iii) If instead $p_3$ does not intersect l, what can be said about $\lambda,\mu$?

(iv) Find a Cartesian equation of the plane containing l and $(1,-1,3)$.

(a) Points P,Q,R have position vectors $\mathbf p,\mathbf q,\mathbf r$. P and Q are fixed; R varies. Given $\mathbf r=(x,y,z)$, $\mathbf p=(-1,2,4)$, $\mathbf q=(3,-5,2)$ and $(\mathbf r-\mathbf p)\cdot\mathbf q=0$, find the relation between x,y,z and describe the locus of R.

(b) Interpret $\mathbf r\cdot\mathbf n=d$ geometrically, where $\mathbf n$ is a constant unit vector and d is a constant scalar. State what d represents.

(c) Given $\mathbf b\cdot\mathbf n\ne0$, solve $\mathbf r=\mathbf a+t\mathbf b$ and $\mathbf r\cdot\mathbf n=d$ for t in terms of $\mathbf a,\mathbf b,\mathbf n,d$. Interpret the solution geometrically.

A right opaque pyramid has square base ABCD at ground level, centre O and vertex V. Perpendicular unit vectors $\mathbf i,\mathbf j,\mathbf k$ point along AB, AD, OV, respectively. AB has length 8 and OV has length 2h. A point light source is at $P(20,-4,0)$. A screen of height 35 has its base on the ground and lies in $\mathbf r\cdot(1,0,0)=\alpha$, where $\alpha<-4$, as shown.

(i) Find a vector equation for the line depicting the light ray from P to V, in terms of h.2 marks

(ii) Find an inequality between $\alpha,h$ so the pyramid’s shadow does not exceed the screen height.3 marks

The point light is now replaced by parallel light rays perpendicular to the screen; also $h=10$.

(iii) Find the exact length of the shadow cast by edge VB on the screen.3 marks

A mirror is placed on plane VBC.

(iv) Find a vector equation of plane VBC and hence the mirror’s inclination to the ground.4 marks