Practice section 5 of 7

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]

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

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]

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]

Planes $\pi_1,\pi_2$ have equations $\mathbf r\cdot(0,1,2)=2$, $\mathbf r\cdot(2,-1,4)=12$. A=(7,2,0), O is the origin.

(i) Verify A lies in both planes. [1]

(ii) They meet in line L. Show that a vector equation of L is $\mathbf r=(7,2,0)+\lambda(3,2,-1)$, $\lambda\in\mathbb R$. [2]

(iii) B=(7,−4,9), and N is its perpendicular foot onto $\pi_2$. Find N. [3]

(iv) Find the Cartesian equation of $\pi_3$ containing L and perpendicular to $\pi_2$. [3]

Prism ABCDEF has rectangular base ABEF. Triangles ABC, DEF are congruent and perpendicular to the base. A,C have position vectors $(−1,3,−5)$, $(3,7,5)$. The unit vector in the direction FE is $(5,−3,2)/\sqrt{38}$ and BC has equation $\mathbf r=(6,2,2)+\lambda(-3,5,3)$, λ real. Find:

(i) a vector equation of AB and the position vector of B; [3]

(ii) an equation of plane ABC in the form $\mathbf r\cdot\mathbf n=D$; [3]

(iii) a vector equation of BE; [2]

(iv) the position vector of the perpendicular foot from C onto plane ABEF. [4]

Plane Π has equation $\mathbf r\cdot(1,2,-3)=5$. Line l passes through P with position vector $(7,4,−6)$ and is parallel to $(7,3,−5)$.

(i) Find the perpendicular distance from the origin to Π. [3]

(ii) Find the acute angle between l and Π. [3]

(iii) Find the position vector of the perpendicular foot from P to Π. Hence find the position vector of P’s reflection in Π. [6]

Planes $\Pi_1,\Pi_2$ are $\mathbf r\cdot(2,4,1)=10$, $\mathbf r\cdot(1,3,1)=8$.

(i) Find their acute angle. [2]

(ii) Obtain a vector equation of their intersection line $l_1$. [2]

Another line $l_2$ is $(x-2)/6=(7-z)/3$, y=m, where m is real.

(iii) If Π1 and l2 intersect at (6,m,5), find m. [2]

(iv) Show l1 and l2 are perpendicular for all m. [2]

Line l is $x-a=y+2=(z+5)/5$, and plane p1 is $3x+2y-z=4$. Plane p1 contains l.

(i) Show a=1. [2]

Plane p2 passes through E=(−2,−5,8) and meets p1 in l.

(ii) Find the Cartesian equation of p2. [3]

(iii) Find N, the perpendicular foot from E to p1. [3]

Plane p3 contains E,N and does not intersect l.

(iv) Find its Cartesian equation. Hence find the distance from l to p3. [4]