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An AGJ is an annular gap junction - a moon-shaped intercellular connection between certain animal cell types.
An AGJ is an annular gap junction - a moon-shaped intercellular connection between certain animal cell types.
answer .com is un agj;gh
One way to answer this question is from first principles. Let
q = a + bi + cj + dk
and
v = e + fi + gj + hk
Then,
qv
= ae - bf - cg - dh
+ i(af + be + ch - dg)
+ j(ag - bh + ce + df)
+ k(ah + bg - cf + de)
= ae - bf - cg - dh + afi + bei + agj +cej + ahk + dek
+ i(ch - dg)
+ j(df - bh)
+ k(bg - cf)
and (keeping in mind that reals commute under multiplication)
vq
= ae - bf - cg - dh
+ i(af + be - ch + dg)
+ j(ag + bh + ce - df)
+ k(ah - bg + cf + de)
= ae - bf - cg - dh + afi + bei + agj + cej + ahk + dek
- i(ch-dg)
- j(df-bh)
- k(bg-cf)
Let m = ae - bf - cg - dh + afi + bei + agj +cej + ahk + dek and n = i(ch - dg) + j(df - bh) + k(bg - cf). Then,
qv = m + n
and
vq = m - n.
For qv to equal vq, we must have n = 0. For n to equal zero, each of its parts must equal zero also, that is,
ch = dg
df = bh
bg = cf
so that either
c/g = d/h
d/h = b/f
b/f = c/g
i.e. (bi + cj + dk) is a multiple (not necessarily integer) of (fi + gj + hk)
or
b = 0, c = 0, d = 0,
or
f = 0, g = 0, h = 0.
Therefore, the nonreal part of v must be a multiple (possibly zero) of the nonreal part of q. More concisely, quaternions q and v commute under multiplication if and only if
v = pq + r
where p and r are real.
A neater proof, requiring more background, can also be given.
Quaternions are often regarded as the combination of a scalar part (the real part of the quaternion) and the vector part (the part containing i, j, and k.) That is, the quaternion a + bi + cj + dk corresponds to the combination of the scalar a and the vector
qv = ac + ad + bc - b·d + b×d.
In the product qv, the order of the factors in each of these five products is reversed. That is, the product is now
vq = ca + da + cb - d·b + d×b.
The first four products commute, whereas the last (a cross product) does not. Therefore,
vq = ac + ad + bc - b·d + d×b.
so that qv = vq if and only if b×d = d×b. It is known that the cross product is anticommutative, meaning that b×d = -d×b. Therefore, this condition reduces to b×d = θ, the zero vector, meaning that one is a scalar multiple of the other (i.e. they lie along the same line.) This gives the same result as the purely algebraic derivation above.
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TOTALLLY AWESUM DOOOOOOOOOOOOOOOOOOOOD
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-6h0 1s,G 7ng -6nf 29,G 7eg -6na 1p,G 72i -6rc 22,G 788 -6mj m,G 7eg -6os 17,G 7ga -6mc 1p,G 7n6 -6h2 1i,B 868 -6ch b,B 83r -6c7 22,B 81i -6eb 1j,B 80d -6cr 1d,B 80s -6hm 1e,B 7qm -6j8 1e,B 7nb -6lo 1f,B 7m3 -6pl 1k,B 7gk -6oi 15,B 7c5 -6su 23,B 72d -70m 1t,B 6ve -738 28,B 70o -71a 1i,B 74j -6u6 1c,B 7b7 -6s7 o,B 7c5 -6qq 1p,B 7gm -6n5 2,B 7jv -6m7 1l,B 7m8 -6iu 1i,B 7qc -6gt 19,B 40c se 2j,B 3ut se 2q,B 3t2 sh 2n,B 3rk rq 3s,B 3pr rq 2q,B 3od qo 3s,B 3mk qc 3c,B 3kr qc 2q,B 3jc q1 3c,B 3hj pa 3g,B 3g5 pa 2q,B 3ec OJ 3l,B 3cj o8 2q,B 3b1 od 2h,B 39f nc 47,B 386 ma 3o,B 36c mb 2k,B 34j mc 2p,B 333 MD 2q,B 317 ko 45,B 2vk kl 2q,B 2tl kp 2k,B 2rq ke 38,B 2qc k3 38,B 2os k9 2h,B 2mq IQ 46,B 2le h8 4d,B 2jk h4 2q,B 2hv hl 22,B 2ga g5 47,B 2er g4 2u,B 2df fr 33,B 2c1 fq 2q,B 2aj fp 2s,B 294 fc 33,B 277 f2 33,B 25m ek 3b,B 245 ea 32,B 22c dq 3b,B 20u de 33,B 1vh d6 33,B 1ts cp 37,B 1s2 c9 39,B 1qj c2 30,B 1oq c2 2q,B 1n3 at 3s,B 1ll af 3c,B 1jp a7 2r,B 1i6 9s 35,B 1gd 9g 32,B 1ev 9c 2s,B 1dk 8o 3g,B 1cb 84 3l,B 1al 81 2s,B 15l 6n 3i,B 13k 5r 3l,B 114 53 3c,B 137 72 39,B 10s 69 3k,B v3 4h 46,B uo 4h 8m,B s1 5f 3e,B PC 3o 3o,G 3af l3 7n,G 3fu mg 7p,G 3jc mn 7i,G 40t s1 5u,G 3t7 q5 4e,G 3om pf 53,G 3j5 or 53,G 3er n1 4l,G 395 m1 4s,G 36g la 66,G 311 l3 5r,G 2tt kc 61,G 2qh j9 60,G 2nl io 6s,G 2mi il 6k,G 2l5 hd 6o,G 2ej fu 6f,G 2i3 h3 83,G 2gp gk 6r,G 2bk g3 71,G 28s ej 72,G 29t er 73,G 2cn fp 6r,G 2bc ee 6k,G 26f eh 6h,G 252 do 6f,G 23i dv 6r,G 21f cv 6o,G 1v9 cv 6d,G 1jk 9m 6p,G 1nc ba 79,G 1pn a7 6u,G 1pn bu 6l,G 1tk bs 6v,G 1sp CD 71,G 1m2 ak 62,G 1lo 9c 6h,G 1i2 9m 47,G 1dh 8t 6j,G 1bv 7n 6i,G 19a 7l 64,G 17o 6k 64,G 17j 6k 6f,G 15b 6d 5n,G 148 52 68,G vk 3v 6a,G 5na -1dv 2h,G 5ja -1c5 2m,G 5hf -18s 2g,G 5l2 -152 18,G 5hm -133 2c,G 5jk -v4 1l,G 5ge -tb 2k,G 5en -qv 34,G 5e6 -n2 2j,G 5cp -jd 21,G 59f -if 38,G 55q -hh 31,G 512 -g2 36,G 4ur -cc 2o,G 4ta -8i 3g,G 4q8 -5l 3k,G 4ph -b 3i,G 4of 1o 3b,G 4ls 6p 49,G 4o9 2m 4h,G 4qa -14 4l,G 4v0 -1b 46,G 4sr -69 46,G 4v8 -7h 4a,G 52j -7r 3h,G 52k -7a 3d,G 55b -37 2r,G 522 -16 37,G 50j 1t 32,G 50l 2f 3o,G 4u8 3d 3m,G 4vi 4j 3j,G 501 56 3q,G 4st 62 3i,G 4t2 7m 3r,G 4rj 8n 3k,G 4q8 9e 3l,G 4om a2 41,G 4nr bp 4c,G 4lv cc 4a,G 4ka e1 43,G 4k7 ed 4b,G 4ka gh 4k,G 4ic fl 3u,G 4gv g2 4b,G 4e4 j3 3t,G 4eh hh 4t,G 4fa h5 4o,G 47j lr 46,G 49e m7 4b,G 4ah la 4q,G 4a5 j9 4i,G 4b1 ji 4e,G 4ar hn 3d,G 481 kc 2l,G 47d lp 43,G 432 n8 3m,G 40i qc 36,G 3u1 su 34,G 3nq 103 2s,G 3l7 139 3r,G 3h8 13b 3s,G 3e8 14j 3h,G 3cp 15u 3u,G 399 16f 3c,G 36k 16i 4e,G 35q 174 4f,G 336 17b 4d,G 2vt 17l 47,G 2sp 18r 55,G 2pu 18g 5f,G 2m1 18o 4m,G 2i5 181 5r,G 2ep 173 63,G 29f 16v 5g,G 23t 15s 44,G 1vd 12v 65,G 1r5 10d 69,G 1mu td 6h,G 1l5 qa 6u,G 1h8 MD 6q,G 1eg kt 69,G 1e7 hh 6s,G 1ak du 70,G 19t eg 7f,G 170 ch 75,G 12n 9t 7d,G 11m 91 6u,G v2 6m 6u,G ta 5o 7d,G r0 3s 78,G pl 2r 79,G l1 35 5j,G hc 1t 6h,G eq 1e 6b,G d7 19 55,G 96 m 4k,B 5pb -1hc i,B 5og -1fu r,B 5o7 -1ep b8,B 5o7 -1db b8,B 5nt -1bk e,B 5nk -1a6 b8,B 5n2 -18o e,B 5mg -171 b,B 5lb -15j 15,B 5kf -146 r,B 5k6 -131 i,B 5k6 -11j b8,B 5jt -10e i,B 5jk -v0 e,B 5jb -ti b8,B 5if -s5 r,B 5ht -r0 r,B 5h1 -pi r,B 5gf -o4 i,B 5fs -ml q,B 5f1 -l7 15,B 5ef -k3 12,B 5ds -il e,B 5da -hg 12,B 5c5 -et l,B 5ba -df r,B 5an -c2 r,B 5a5 -ak r,B 599 -9f 15,B 57r -81 1d,B 579 -6s i,B 564 -4a u,B 54m -2s 1d,B 538 -1e 1d,B 52m 0 e,B 52m 1e b8,B 524 2j r,B 50p 41 1a,B 4vb 5f 1d,B 4u6 6k 1d,B 4t1 7p 1d,B 4rj 97 1d,B 4qf ab 1d,B 4p1 bp 1d,B 4ns cu 1d,B 4me ec 1d,B 4l9 fi 1c,B 4js gv 1d,B 4ie hi 24,B 4h9 in 1d,B 4fr k4 1d,B 4ed li 1d,B 4d8 ls 2c,B 4bq m5 2c,B 4ad NJ 1d,B 48v o5 2c,B 47h on 1o,B 45q q5 1j,B 44l qo 1v,B 43g ra 1o,B 41p t1 1d,B 40k u6 1d,B 3vg v1 1o,B 3u2 vt 1o,B 3sk 10f 1o,B 3rf 112 28,B 3q1 11k 1v,B 3ot 11t 2c,B 3nf 12p 1l,B 3m1 13b 2c,B 3ks 146 1l,B 3jn 15b 1d,B 3ij 15b 2q,B 3h5 15k 2c,B 3fn 16g 1v,B 3e9 16g 2q,B 3cr 16p 2c,B 3bm 17l 1l,B 3a8 17l 2q,B 392 17j 2t,B 386 18e 1d,B 36m 199 2g,B 35h 199 2q,B 340 19g 2o,B 32v 19k 2k,B 31p 19t 2a,B 30j 1a3 2d,B 2v7 1a7 2q,B 2u2 1a1 3b,B 2sk 1ag 1o,B 2rc 1an 2e,B 2qa 1ar 2q,B 2ov 1at 2l,B 2nq 1at 2q,B 2ml 1au 2o,B 2l5 1b0 2n,B 2k0 1b0 2q,B 2ir 1ar 31,B 2hi 1ag 3c,B 2g4 1ac 2t,B 2ek 1ab 2q,B 2dd 1a8 2u,B 2c5 1a7 2s,B 2ar 1a1 34,B 297 19s 2q,B 281 19p 2u,B 26m 191 3l,B 25s 18d 41,B 24e 180 2m,B 22t 17n 34,B 21r 17l 2v,B 1vm 16e 46,B 1up 15q 3t,B 1tf 154 3n,B 1s8 149 3q,B 1qk 13a 41,B 1pf 12e 41,B 1np 11b 3t,B 1mk 107 46,B 1l6 uv 45,B 1kb u8 42,B 1iv st 47,B 1hp rp 47,B 1gp qh 4f,B 1fq PC 4c,B 1f2 o9 4g,B 1e2 mr 4h,B 1co ld 48,B 1bs k7 4i,B 1a6 ig 47,B 199 hl 46,B 17r g7 47,B 16d ep 47,B 150 db 47,B 13r c6 47,B 12m b1 47,B 118 9k 47,B vq 86 47,B um 72 46,B t8 5l 46,B rr 4i 43,B qm 49 38,B pb 3v 39,B o6 38 42,B mo 32 30,B la 2r 35,B k5 2m 31,B ij 2i 2u,B gs 1v 38,B fh 1l 31,B e3 1d 34,B cg 1a 2q,B av 17 2u,B 9q 12 33,B 8b r 32,B 6k r 2q
