VECTOR ALGEBRA
of vector plots for potential differences and currents (they could all be on the
same plot if suitable scales were chosen), determine all unknown currents and
potential differences and find values for the inductance ofLand the resistance
ofR 2.
[ Scales of 1 cm = 0.1 V for potential differences and 1 cm = 1 mA for currents
are convenient. ]7.11 Hints and answers7.1 (c), (d) and (e).
7.3 (a) A sphere of radiuskcentred on the origin; (b) a plane with its normal in the
direction ofuand at a distancelfrom the origin; (c) a cone with its axis parallel
touand of semiangle cos−^1 m; (d) a circular cylinder of radiusnwith its axis
parallel tou.
7.5 (a) cos−^1
√
2 /3; (b)z−x=2;(c)1/√
2; (d)^1312 (c×g)·j=^13.
7.7 Show thatq×ris parallel top; volume =^13
[ 1
2 (q×r)·p]
=^53.
7.9 Note that (a×b)·(c×d)=d·[(a×b)×c] and use the result for a triple vector
product to expand the expression in square brackets.
7.11 Show that the position vectors of the points are linearly dependent;r=a+λb
wherea=i+kandb=−j+k.
7.13 Show thatpmust have the directionˆn×mˆand writeaasxnˆ+ymˆ. By obtaining a
pair of simultaneous equations forxandy, prove thatx=(λ−μnˆ·mˆ)/[1−(nˆ·mˆ)^2 ]
and thaty=(μ−λˆn·mˆ)/[1−(nˆ·mˆ)^2 ].
7.15 (a) Note that|a−g|^2 =R^2 =| 0 −g|^2 , leading toa·a=2a·g.
(b) Makepthe new origin and solve the three simultaneous linear equations to
obtainλ=5/18,μ=10/18,ν=− 3 /18, givingg=2i−kand a sphere of
radius
√
5 centred on (5, 1 ,−3).
7.17 (a) Find two points on both planes, say (0, 0 ,0) and (1,− 2 ,1), and hence determine
the direction cosines of the line of intersection; (b) (^23 )^1 /^2.
7.19 For (c) and (d), treat (c×a)×(a×b) as a triple vector product withc×aas one
of the three vectors.
7.21 (b)b′=a−^1 (−i+j+k),c′=a−^1 (i−j+k),d′=a−^1 (i+j−k); (c)a/2fordirection
k; successive planes through (0, 0 ,0) and (a/ 2 , 0 ,a/2) give a spacing ofa/
√
8for
directioni+j; successive planes through (−a/ 2 , 0 ,0) and (a/ 2 , 0 ,0) give a spacing
ofa/√
3fordirectioni+j+k.
7.23 Note thata^2 −(ˆn·a)^2 =a^2 ⊥.
7.25 p=− 2 a+3b,q=^32 a−^32 b+^52 candr=2a−b−c. Remember thata·a=b·b=
c·c=2anda·b=a·c=b·c=1.
7.27 With currents in mA and potential differences in volts:
I 1 =(7. 76 ,− 23. 2 ◦),I 2 =(14. 36 ,− 50. 8 ◦),I 3 =(8. 30 , 103. 4 ◦);
V 1 =(0. 388 ,− 23. 2 ◦),V 2 =(0. 287 ,− 50. 8 ◦),V 4 =(0. 596 , 39. 2 ◦);
L= 33 mH,R 2 =20Ω.