Practice section 6 of 7

(a) Plane Π passes through A=(2,0,0) parallel to $-2\mathbf i+\mathbf j+\mathbf k$ and $\mathbf i-3\mathbf j+2\mathbf k$.

(i) Find Π in scalar-product form and verify B=(6,−5,1) lies on it. [3]

(ii) Find the position vector of the perpendicular foot from C=(−2,1,−3) to Π. [4] Hence find the equation of a parallel plane Π1 such that C is equidistant from Π and Π1. [2]

(b) Line l and plane P are $l:\mathbf r=\lambda(\mathbf u+\mathbf v)$, λ real, and $P:\mathbf r\cdot(\mathbf u\times\mathbf v)=0$, with u,v nonzero and nonparallel.

(i) Show l is contained in P. [2]

(ii) For u=(−1,2,1), v=(3,−3,2), find a line in P perpendicular to l through the origin. [2]

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]

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

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

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

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]

Use b=1 for the rest.

(ii) Find N, their intersection. [3]

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

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

(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]

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]

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

Plane Π2 contains B and l1.

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

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

(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]

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]

(ii) Show AB lies in π1. [1]

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

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

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

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]

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

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] Hence deduce, when α>0, the Cartesian equation of π2. [3]