Review practice · Section 2 of 3

Planes p1,p2 are $2x-5y+3z=3$ and $3x+\lambda y+6z=\mu$.

(i) They intersect in l:$\mathbf r=(4,-2,-5)+s(-2,1,3)$, s real. Show λ=−12 and find μ.3 marks

Plane p3 is $5x+8y+tz=12$.

(ii) Using(i), find the exact t if p3 does not intersect l.2 marks

(iii) Plane p4 contains l and (1,−1,2). Find its Cartesian equation.3 marks

A has position vector $-\mathbf i-11\mathbf j+2\mathbf k$. Line l is $(x-9)/7=y+3=-(z+1)/2$, and plane p is $5x+3y+bz+1=0$.

(i) If l and p meet in exactly one point, what can be said about b?2 marks

Use b=1 for the rest.

(ii) Find N, their intersection.3 marks

(iii) Show the shortest distance from A to p is $\sqrt{35}$.3 marks

(iv) Plane p1 is parallel to p at shortest distance $2\sqrt{35}$. Find a possible equation in scalar-product form.3 marks

(v) Plane p undergoes, in order: translation 3 units in negative x; scaling parallel to the y-axis by factor 1/2; reflection in the xy-plane. Find a Cartesian equation of the resulting p2.3 marks

A,B have position vectors $\mathbf i-2\mathbf j-\mathbf k$ and $-\mathbf i-2\mathbf k$. Line l1 goes through A parallel to $5\mathbf i+2\mathbf j-\mathbf k$. Plane Π1 is $\mathbf r\cdot(-1,1,2)=3$.

(i) Find the acute angle between l1 and Π1.2 marks

(ii) Find the position vector of the perpendicular foot N from B to Π1. Hence find the reflected point B′.4 marks

Plane Π2 contains B and l1.

(iii) Find Π2 in scalar-product form.3 marks

(iv) Find the Cartesian equation of intersection l2 of Π1,Π2.2 marks

(v) Π3 is the reflection of Π2 in Π1. Find an equation $\mathbf r=\mathbf u+s\mathbf p+t\mathbf q$, with s,t real.2 marks

A=(0,2,0), B=(1,0,2), and plane π1 is $\mathbf r\cdot(2,2,1)=4$.

(i) Find a vector equation of AB.2 marks

(ii) Show AB lies in π1.1 mark

(iii) Plane π2 contains AB and is perpendicular to π1. Show its equation is $\mathbf r\cdot(-2,1,2)=2$.3 marks

(iv) P=(p,0,0) is equidistant from π1,π2. Calculate p.5 marks

(v) Hence write down a line equidistant from π1,π2 which does not intersect either plane.2 marks

P has position vector $2\mathbf i+\mathbf j-2\mathbf k$ and lies on $\ell:(1-y)/3=(z+2)/4$, x=2. Plane π1 is $x-2y-2z=8$.

(i) Show the intersection Q of ℓ,π1 is (2,7,−10).3 marks

(ii) Find the exact sine of the acute angle between ℓ,π1. Hence find the exact length of projection of PQ onto π1.4 marks

Plane π2 is parallel to π1. M has position vector $2\mathbf i+(4\alpha+4)\mathbf j+(3\alpha-6)\mathbf k$ and is equidistant from ℓ and π2.

(iii) Find α if the distance from M to ℓ is 10. 3 marks Hence deduce, when α>0, the Cartesian equation of π2.3 marks

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 marks

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

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 marks