VECTOR ALGEBRA
7.12 The planeP 1 contains the pointsA,BandC, which have position vectors
a=− 3 i+2j,b=7i+2jandc=2i+3j+2k, respectively. PlaneP 2 passes
throughAand is orthogonal to the lineBC, whilst planeP 3 passes through
Band is orthogonal to the lineAC. Find the coordinates ofr, the point of
intersection of the three planes.
7.13 Two planes have non-parallel unit normalsnˆandmˆand their closest distances
from the origin areλandμ, respectively. Find the vector equation of their line
of intersection in the formr=νp+a.
7.14 Two fixed points,AandB, in three-dimensional space have position vectorsa
andb. Identify the planePgiven by
(a−b)·r=^12 (a^2 −b^2 ),
whereaandbare the magnitudes ofaandb.
Show also that the equation
(a−r)·(b−r)=0
describes a sphereSof radius|a−b|/2. Deduce that the intersection ofPandS
is also the intersection of two spheres, centred onAandB, and each of radius
|a−b|/
√
2.
7.15 LetO,A,BandCbe four points with position vectors 0 ,a,bandc, and denote
byg=λa+μb+νcthe position of the centre of the sphere on which they all lie.
(a) Prove thatλ,μandνsimultaneously satisfy
(a·a)λ+(a·b)μ+(a·c)ν=^12 a^2
and two other similar equations.
(b) By making a change of origin, find the centre and radius of the sphere on
which the pointsp=3i+j− 2 k,q=4i+3j− 3 k,r=7i− 3 kands=6i+j−k
all lie.
7.16 The vectorsa,bandcare coplanar and related by
λa+μb+νc=0,
whereλ,μ,νare not all zero. Show that the condition for the points with position
vectorsαa,βbandγcto be collinear is
λ
α+
μ
β+
ν
γ=0.
7.17 Using vector methods:
(a) Show that the line of intersection of the planesx+2y+3z=0and
3 x+2y+z= 0 is equally inclined to thex-andz-axes and makes an angle
cos−^1 (− 2 /√
6) with they-axis.
(b) Find the perpendicular distance between one corner of a unit cube and the
major diagonal not passing through it.7.18 Four pointsXi,i=1, 2 , 3 ,4, taken for simplicity as alllying within the octant
x, y, z≥0, have position vectorsxi. Convince yourself that the direction of
vectorxnlies within the sector of space defined by the directions of the other
three vectors if
min
overj[
xi·xj
|xi||xj|]
,
considered fori=1, 2 , 3 ,4 in turn, takes its maximum value fori=n,i.e.nequals
that value ofifor which the largest of the set of angles whichximakes with
the other vectors, is found to be the lowest. Determine whether any of the four