Review practice · Section 1 of 3

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 mark

(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 marks

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

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

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 marks

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

(iii) a vector equation of BE;2 marks

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

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 marks

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

(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 marks

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 marks

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

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 marks

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

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 marks

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

(ii) Find the Cartesian equation of p2.3 marks

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

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

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

(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 marks

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

(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 marks

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