Practice section 7 of 7

A=(4,1,0), and line l1 is $x-3=-y=(z-3)/2$.

(i) Find plane π containing A and perpendicular to l1, in scalar-product form. [3]

(ii) Find P on l1 with AP perpendicular to l1. [3]

A ray travels along l2 from (0,−9,1) to P and reflects in π along l3, in the same plane as l1,l2, with equal angles between l1,l2 and l1,l3. Find l3. [6]

Line l is $x-1=(y+2)/(-1)=(z-1)/2$, and plane p1 has equation $\mathbf r=\alpha\mathbf i+(-\alpha+\beta)\mathbf j+(12+2\alpha-5\beta)\mathbf k$, α,β real.

(i) Show l is parallel to p1. [1]

(ii) A,B vary independently on l,p1. C lies on segment AB with BC=2AC. Show C lies on a plane as A,B vary. [3]

(iii) Calling this plane p2, show its equation is $\mathbf r\cdot(3,5,1)=0$. [2]

(iv) Find the perpendicular distance between l,p2. Hence or otherwise find the distance between p1,p2. [3]

The closed cuboid has OA = 12 cm, OC = 10 cm, OE = 5 cm. O is the origin; perpendicular unit vectors i,j,k are parallel to OA, OC, OE. Vertex order is as in the diagram.

A yellow ray starts at O in direction $\mathbf i+2\mathbf j+2\mathbf k$. A blue ray starts at A and passes through H on ED with EH:ED=2:5.

(i) Will the yellow and blue paths cross? Explain clearly. [4]

(ii) Find the angle between the blue ray and plane DEFG. [2]

(iii) Find X where the yellow ray meets DEFG. [3]

(iv) DEFG is reflective. The yellow ray reflects and meets OABC at Y. Find the area of triangle OXY. [3]

A laser source at O projects an incident beam with direction $\mathbf i+2\mathbf j+\mathbf k$. It hits a mirror at P with angle θ and reflects at the same angle. The mirror is $-x+2y+3z=12$.

(i) Find the acute angle θ the beam makes with the mirror.

(ii) By finding O′, the image of O in the mirror, find a vector equation of the line containing the reflected beam.

(iii) A sensor at $(3,1,0)$ must be less than 2 units from either the incident or reflected beam to work properly. Determine whether it will work.