{"id":6261,"date":"2024-08-14T08:01:06","date_gmt":"2024-08-14T00:01:06","guid":{"rendered":""},"modified":"2024-08-14T08:01:06","modified_gmt":"2024-08-14T00:01:06","slug":"\u5411\u91cf\u70b9\u79ef\u548c\u53c9\u4e58_\u5411\u91cf\u53c9\u4e58\u518d\u70b9\u4e58\u600e\u4e48\u8fd0\u7b97","status":"publish","type":"post","link":"https:\/\/mushiming.com\/6261.html","title":{"rendered":"\u5411\u91cf\u70b9\u79ef\u548c\u53c9\u4e58_\u5411\u91cf\u53c9\u4e58\u518d\u70b9\u4e58\u600e\u4e48\u8fd0\u7b97"},"content":{"rendered":"

\u53c2\u8003\uff1a\u300a\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u300b
\u53c2\u8003\uff1a \u767e\u5ea6\u767e\u79d1
\u53c2\u8003\uff1a http:\/\/zuoye.baidu.com\/question\/f78a7e9b076367b03f1df832a8c131b3.html
\u53c2\u8003\uff1a https:\/\/zh.wikipedia.org\/wiki\/%E6%AD%A3%E5%BC%A6%E5%AE%9A%E7%90%86
\u53c2\u8003\uff1ahttps:\/\/zh.wikipedia.org\/wiki\/%E7%82%B9%E7%A7%AF
\u53c2\u8003\uff1ahttps:\/\/zh.wikipedia.org\/wiki\/%E5%8F%89%E7%A7%AF<\/p>\n

(Owed by: \u6625\u591c\u559c\u96e8 http:\/\/blog.csdn.net\/chunyexiyu \u8f6c\u8f7d\u8bf7\u6807\u660e\u6765\u6e90)<\/p>\n

\u70b9\u79ef\/\u70b9\u4e58\u5e94\u7528\uff1a<\/strong>
\u7269\u7406\u5b66\u4e2d\u529b\u5b66\u7684\u529b\u505a\u529f\u7684\u95ee\u9898\uff0c\u7ecf\u5e38\u7528\u5230\u70b9\u79ef\u8ba1\u7b97\u3002
\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u5e38\u7528\u6765\u8fdb\u884c\u65b9\u5411\u6027\u5224\u65ad\uff0c\u5982\u4e24\u5411\u91cf\u70b9\u79ef\u5927\u4e8e0\uff0c\u5219\u5b83\u4eec\u7684\u65b9\u5411\u671d\u5411\u76f8\u8fd1\uff1b\u5982\u679c\u5c0f\u4e8e0\uff0c\u5219\u65b9\u5411\u76f8\u53cd\u3002<\/p>\n

\u53c9\u79ef\/\u53c9\u4e58\u5e94\u7528\uff1a<\/strong>
\u5728\u7269\u7406\u5b66\u529b\u5b66\u3001\u7535\u78c1\u5b66\u3001\u5149\u5b66\u548c\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u7b49\u7406\u5de5\u5b66\u79d1\u4e2d\uff0c\u53c9\u79ef\u5e94\u7528\u5341\u5206\u5e7f\u6cdb\u3002\u4f8b\u5982\u529b\u77e9\u3001\u89d2\u52a8\u91cf\u3001\u6d1b\u4f26\u5179\u529b\u7b49\u77e2\u91cf\u90fd\u53ef\u4ee5\u7531\u5411\u91cf\u7684\u53c9\u79ef\u6c42\u89e3\u3002\u5728\u8fdb\u884c\u8fd9\u4e9b\u7269\u7406\u91cf\u7684\u8ba1\u7b97\u65f6\uff0c\u5f80\u5f80\u53ef\u4ee5\u501f\u52a9\u53f3\u624b\u5b9a\u5219\u8f85\u52a9\u5224\u65ad\u65b9\u5411\u3002<\/p>\n

 <\/p>\n

0. \u516c\u5f0f\u63a8\u5012-\u80cc\u666f\u516c\u5f0f<\/h4>\n

\u5706\u5468\u89d2\u5b9a\u7406\uff1a<\/strong>
\u540c\u4e00\u6761\u5f27\u6240\u5bf9\u5706\u5468\u89d2\u7b49\u4e8e\u5b83\u6240\u5bf9\u5706\u5fc3\u89d2\u7684\u4e00\u534a<\/p>\n

\u4f59\u5f26\u5b9a\u7406\uff1a<\/strong>\uff08\u53cd\u6620\u8fb9\u548c\u5bf9\u89d2\u4f59\u5f26\u7684\u5173\u7cfb\uff0c\u4e09\u6761\u8fb9\u53ef\u4ee5\u786e\u5b9a\u89d2\u5ea6\uff0c\u4e8c\u6761\u8fb9\u52a0\u5939\u89d2\u53ef\u4ee5\u786e\u5b9a\u53e6\u4e00\u8fb9\u957f\u5ea6\uff09
\u8bbe\u8fb9\u4e3aABC\uff0c\u8fb9\u5bf9\u89d2\u4e3aabc\uff0c\u5219\uff1a 
A^2 = B^2 + C^2 - 2BCcosa
B^2 = A^2 + C^2 - 2ACcosb
C^2 = A^2 + B^2 - 2ABcosc<\/strong><\/p>\n

\u6b63\u5f26\u5b9a\u7406\uff1a<\/strong>\uff08\u53cd\u6620 \u8fb9\/\u5bf9\u89d2\u6b63\u5f26\u6bd4\u503c \u4e3a\u5916\u63a5\u77e9\u5f62\u7684\u76f4\u5f84\u957f\u5ea6\uff09
\u8bbe\u8fb9\u4e3aABC\uff0c\u8fb9\u5bf9\u89d2\u4e3aabc\uff0c\u5219\uff1a <\/strong>
A\/sina = B\/sinb = C\/sinc = 2R \uff08R\u4e3a\u4e09\u89d2\u5f62\u5bf9\u5e94\u7684\u5916\u5207\u5706\uff0c\u4f7f\u7528\u5706\u5468\u89d2\u5b9a\u7406\u5f88\u5bb9\u6613\u8bc1\u660e\u8be5\u5b9a\u7406\uff09<\/p>\n

1. \u4e8c\u7ef4\u5411\u91cf<\/h4>\n

\u8bbe\u5411\u91cf:<\/strong>
V1(X, Y, Z)    <\/strong>
\u6781\u5750\u6807\u7cfb\u4e0b\u8868\u793a\u5411\u91cf\u4e3a\uff1a
V1(R, cosx, siny) <\/strong>
\u6ce8\uff1a(R-\u5411\u91cf\u957f\u5ea6\uff0ccosx-\u5411\u91cf\u4e0eX\u6b63\u8f74\u5939\u89d2cos\u503c\uff0csinx-\u5411\u91cf\u4e0eX\u6b63\u8f74\u5939\u89d2\u7684sin\u503c)<\/p>\n

\u5219\u5411\u91cf\u7684\u957f\u5ea6\u8ba1\u7b97\uff1a<\/strong>
R = \u6839\u53f7(X^2 + Y^2)
\u501f\u7528\u6781\u5750\u6807\u65b9\u5f0f\u53cd\u63a8\u8ba1\u7b97\u4e3a\uff1acos\u89d2\u5ea6\u975e0\u65f6\u652f\u6301
R = |X \/ cosx|
R = |Y \/ sinx|<\/p>\n

\u4ece\u800c\u4e24\u4e2a\u5411\u91cf\u957f\u5ea6\u76f8\u4e58\u65f6\uff1a<\/strong>
R1 * R2 = \u6839\u53f7[(X1^2+Y1^2) * (X2^2+Y2^2)]
\u501f\u7528\u6781\u5750\u6807\u65b9\u5f0f\uff1acos\u89d2\u5ea6\u975e0\u65f6\u652f\u6301
R1 * R2 = |X1\/cosx1| * |X2\/cosx2| = |X1 * X2 \/ cosx1 \/ cosx2|
R1 * R2 = |Y1\/sinx1| * |Y2\/sinx2| = |Y1 * Y2 \/ sinx1 \/ sinx2|<\/p>\n

\u70b9\u79ef\/\u70b9\u4e58DotProduct\uff1a
V1.V2 = x1*x2 + y1*y2 <\/strong>
\u77e9\u9635\u8868\u793a\u4e3a
(x1, y1) * (x2   = x1 * x2 + y1 * y2
                 y2)<\/strong>
\u8bc1\u660e\u8fc7\u7a0b\uff1a
V1.V2 =|V1||V2|cos<V1, V2> = R1R2cos<V1,V2>
V3=R2-R2\uff0c\u56f4\u6210\u4e09\u89d2\u5f62\uff0c\u6839\u636e\u4f59\u5f26\u5b9a\u7406cos<V1,V2> = (R1^2 + R2^2 - R3^2) \/ 2R1R2
\u6240\u4ee5V1.V2 = (R1^2+R2^2-R3^2)\/2 = (x1^2+y1^2+x2^2+y2^2 - (x2-x1)^2 - (y2-y1)^2 ) \/ 2
                  = (2x1x2 + 2y1y2) \/ 2
                  = x1x2 + y1y2
 <\/p>\n

\u53c9\u79ef\/\u53c9\u4e58crossProduct:
cross(V1, V2) = |V1| |V2| sin<V1, V2> * n (n\u6307\u53f3\u624b\u5b9a\u5219\u786e\u5b9a\u51fa\u7684\u5355\u4f4d\u5411\u91cf\uff09
                        = (x1y2 - y1x2) * k (k\u4e3az\u8f74\u5355\u4f4d\u5411\u91cf(0,0,1))
                        = (0, 0, x1y2-y1x2)<\/strong>
\u77e9\u9635\u8868\u793a\u4e3a\uff1ai-X\u8f74\u5355\u4f4d\u5411\u91cf j-Y\u8f74\u5355\u4f4d\u5411\u91cf k-Z\u8f74\u5355\u4f4d\u5411\u91cf   i * j = k   j * i = -K
(i,       j,     k)     =  (y1*0-0*y2,  0*x2-x1*0, x1y2-y1x2) = (0, 0, x1y2-y1x2)
(x1    y1    0)
(x2    y2    0)<\/strong><\/p>\n

\u8bc1\u660e\u8fc7\u7a0b\uff1a<\/strong>
\u53f3\u624b\u5750\u6807\u7cfb\u4e2d\uff0c\u57fa\u5411\u91cfi\uff0cj\uff0ck \u6ee1\u8db3\u4ee5\u4e0b\u7b49\u5f0f\uff1a
i * j = k     j * k = i    k * i = j    j * i = -k    k * j = -i    i * k = -j 
\u53e6\u5916\u53c2\u8003\u5b9a\u4e49\u4e2d\u4e58\u7684sin\u89d2\u5ea6\uff0c\u5219 0 = i * i = j * j = k * k
\u53e6\u5916\u53c9\u4e58\u6ee1\u8db3\u4e58\u6cd5\u7ed3\u5408\u7387\uff0c\u4f7f\u7528i j k\u53d8\u5316V=x * i + y * j
cross(V1, V2) = (x1, y1) (x2, y2)
                       = (x1* i + y1* j) * (x2 * i + y2 * j) 
                       = (x1x2ii + x1y2ij + y1x2ji +  y1y2jj) = x1y2k - y1x2k
                       = (x1y2 - y1x2) k
 <\/p>\n

2. \u4e09\u7ef4\u5411\u91cf<\/h4>\n

\u8bbe\u5411\u91cf:<\/strong>
V(X, Y, Z)
\u6781\u5750\u6807\u7cfb\u4e0b\u8868\u793a\u5411\u91cf\u4e3a\uff1a
V(R, cosx, cosy, cosz)
(cosx-\u5411\u91cf\u4e0eX\u6b63\u8f74\u5939\u89d2cos\u503c\uff0ccosy-\u5411\u91cf\u4e0eY\u6b63\u8f74\u5939\u89d2\u7684cos\u503c, cosz-\u5411\u91cf\u4e0eZ\u6b63\u8f74\u5939\u89d2\u7684cos\u503c)<\/p>\n

\u5219\u5411\u91cf\u957f\u5ea6\uff1a(\u6807\u91cf)<\/strong>
R = \u6839\u53f7(X*X + Y*Y + Z*Z)
\u501f\u7528\u6781\u5750\u6807\u65b9\u5f0f\u53cd\u63a8\u8ba1\u7b97\u4e3a\uff1acos\u89d2\u5ea6\u975e0\u65f6\u652f\u6301
R = |X \/ cosx|
R = |Y \/ cosy|
R = |Z \/ cosZ| <\/strong> <\/p>\n

\u4ece\u800c\u4e24\u4e2a\u5411\u91cf\u957f\u5ea6\u76f8\u4e58\u65f6\uff1a<\/strong>
R1 * R2 = \u6839\u53f7[(X1^2+Y1^2 + Z1^2) * (X2^2+Y2^2+Z2^2)]
\u501f\u7528\u6781\u5750\u6807\u65b9\u5f0f\uff1acos\u89d2\u5ea6\u975e0\u65f6\u652f\u6301
R1 * R2 = |X1\/cosx1| * |X2\/cosx2| * |X3\/cosx3| = |X1 * X2 *X3 \/ cosx1 \/ cosx2 \/ cosx3|<\/p>\n

\u70b9\u79ef\/\u70b9\u4e58DotProduct\uff1a
V1.V2 = x1*x2 + y1*y2 + z1*z2<\/strong>
\u77e9\u9635\u8868\u793a\u4e3a
(x1, y1, z1) * (x2   = x1*x2 + y1*y2 + z1*z2
                      y2
                      z2)<\/strong>
\u8bc1\u660e\u8fc7\u7a0b\uff1a
V1.V2 =|V1||V2|cos<V1, V2> = R1R2cos<V1,V2>
V3=R2-R2\uff0c\u56f4\u6210\u4e09\u89d2\u5f62\uff0c\u6839\u636e\u4f59\u5f26\u5b9a\u7406cos<V1,V2> = (R1^2 + R2^2 - R3^2) \/ 2R1R2
\u6240\u4ee5V1.V2 = (R1^2+R2^2-R3^2)\/2 = (x1^2+y1^2+z1^2+x2^2+y2^2+z2^2 - (x2-x1)^2 - (y2-y1)^2 - (z2-z1)^2) \/ 2
                  = (2x1x2 + 2y1y2 + 2z1z2) \/ 2
                  = x1x2 + y1y2 <\/strong>+ z1z2<\/p>\n

 <\/p>\n

\u53c9\u79ef\/\u53c9\u4e58crossProduct:
cross(V1, V2) = |V1| |V2| sin<V1, V2> * n (n\u6307\u53f3\u624b\u5b9a\u5219\u786e\u5b9a\u51fa\u7684\u5355\u4f4d\u5411\u91cf<\/strong>\uff09
                        = (y1z2-z1y2)i + (z1x2-x1z2)j + (x1y2-y1x2)k (i,j,k\u5206\u522b\u4e3axyz\u8f74\u5355\u4f4d\u5411\u91cf)
                        = (y1z2-z1y2, z1x2-x1z2, x1y2-y1x2)<\/strong>
\u77e9\u9635\u8868\u793a\u4e3a\uff1ai-X\u8f74\u5355\u4f4d\u5411\u91cf j-Y\u8f74\u5355\u4f4d\u5411\u91cf k-Z\u8f74\u5355\u4f4d\u5411\u91cf   i * j = k   j * i = -K
(i,       j,     k)     =  <\/strong>(y1z2-z1y2)i + (z1x2-x1z2)j + (x1y2-y1x2)k  = (y1*z2-z1*y2,  z1*x2-x1*z2, x1y2-y1x2)
(x1    y1    z1)
(x2    y2    z2)<\/strong>
\u6ce8\uff1an<\/strong>\u65b9\u5411\u4e3aV1\u5230V2\u53f3\u624b\u5b9a\u5f8b\u6307\u5411\u65b9\u5411\uff0c\u4e3a\u5782\u76f4\u4e8eV1,V2\u5e73\u9762\u7684\u8f74<\/p>\n

\u8bc1\u660e\u8fc7\u7a0b\uff1a<\/strong>
\u53f3\u624b\u5750\u6807\u7cfb\u4e2d\uff0c\u57fa\u5411\u91cfi\uff0cj\uff0ck \u6ee1\u8db3\u4ee5\u4e0b\u7b49\u5f0f\uff1a
i * j = k     j * k = i    k * i = j    j * i = -k    k * j = -i    i * k = -j 
\u53e6\u5916\u53c2\u8003\u5b9a\u4e49\u4e2d\u4e58\u7684sin\u89d2\u5ea6\uff0c\u5219 0 = i * i = j * j = k * k
\u53e6\u5916\u53c9\u4e58\u6ee1\u8db3\u4e58\u6cd5\u7ed3\u5408\u7387\uff0c\u4f7f\u7528i j k\u53d8\u5316V=x * i + y * j
cross(V1, V2) = (x1, y1, z1) (x2, y2, z2)
                       = (x1*i + y1*j + z1*k) * (x2*i + y2*j + z2*k) 
                       = x1x2ii + x1y2ij + x1z2ik + y1x2ji + y1y2jj + y1z2jk + z1x2ki + z1y2kj + z1z2kk
                       = x1y2k - x1z2j - y1x2k + y1z2i + z1x2j - z1y2i
                       = (y1z2-z1y2)i + (z1x2-x1z2)j + (x1y2-y1x2)k 
                       = (y1z2-z1y2, z1x2-x1z2, x1y2-y1x2)<\/p>\n

 <\/p>\n

3. \u5e94\u7528\u4ecb\u7ecd<\/strong>\uff1a<\/strong><\/h4>\n

\u70b9\u4e58dotProduct\u4e0ecrossProduct\u4e24\u4e2a\u4e4b\u95f4\u7684\u5dee\u522b\u662f\u6bd4\u8f83\u5927\u7684\u3002<\/p>\n

dotProduct\u8ba1\u7b97\u51fa\u7684\u4e00\u4e2a\u6570\u503c\u7ed3\u679c\uff0c\u7c7b\u4f3c\u4e8e\u529f = F * S\uff0c\u6211\u4eec\u77e5\u9053\u540c\u6837\u7684\u529b\u91cf\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u4e8e\u62d4\u6cb3\u628a\u5bf9\u65b9\u62d4\u8fc7\u6765\uff0c\u4e5f\u53ef\u80fd\u88ab\u62d4\u8fc7\u53bb\uff0c\u6548\u679c\u662f\u5b8c\u5168\u4e0d\u4e00\u6837\u7684\uff0c\u8fd9\u5c31\u662f\u529b\u7684\u65b9\u5411\u548c\u79fb\u52a8\u65b9\u5411\u6295\u5f71\u4e00\u81f4\u4e0e\u4e0d\u4e00\u81f4\u7684\u533a\u522b\uff0c\u5bfc\u81f4\u505a\u6b63\u529f\u4e0e\u505a\u8d1f\u529f\u7684\u533a\u522b\u3002
F, S\u65b9\u5411<90\u5ea6\uff0c\u505a\u6b63\u529f
F,S\u5782\u76f4\uff0c\u672a\u505a\u529f
F,S\u65b9\u5411>90\u5ea6\uff0c\u505a\u8d1f\u529f<\/p>\n

\u516c\u5f0f:
dotProduct<\u5411\u91cfA, \u5411\u91cfB> = |A| |B| cos<A, B>
\/\/ \u628a\u529b\u62c6\u6210\u4e09\u4e2a\u65b9\u5411\uff0c\u5206\u522b\u662fx\/y\/z\u4e09\u4e2a\u65b9\u5411\uff0c\u62c6\u6210\u4e09\u4e2a\u65b9\u5411\u540e\uff0c\u56e0\u4e3a\u4e92\u76f8\u5782\u76f4\uff0c\u4e0d\u540c\u65b9\u5411\u7684cos\u4e3a0\uff0c\u6240\u4ee5\u53ea\u6709\u540c\u65b9\u5411\u7684\u624d\u9700\u8981\u8ba1\u7b97
=|Ax| |Bx| + |Ay| |By| + |Ax| |Cz|
= Xa*Xb + Ya*Yb + Za*Zb<\/p>\n

\u8bf4\u660e:
|A| A\u7684\u957f\u5ea6 |B|B\u7684\u957f\u5ea6
\u5411\u91cfA(Xa, Ya, Za)
\u5411\u91cfB(Xb, Yb, Zb)<\/p>\n

\u4e24\u4e2a\u5411\u91cf\u7684\u5939\u89d2<90\u5ea6\u65f6\uff0c\u4e24\u4e2a\u76f8\u91cf\u7ed3\u679c\u4e3a\u6b63\u503c\uff1b
\u5939\u89d2=90\u5ea6\u65f6\uff0c\u7ed3\u679c\u4e3a0
\u5939\u89d2>90\u5ea6\u65f6\uff0c\u7ed3\u679c\u4e3a\u8d1f\u503c<\/p>\n

 <\/p>\n

crossProduct\u8ba1\u7b97\u51fa\u7684\u4e00\u4e2a\u65b0\u5411\u91cf\uff0c\u65b0\u5411\u91cf\u5782\u76f4\u4e8e\u8fd9\u4e24\u4e2a\u8ba1\u7b97\u5411\u91cf\uff0c\u7b26\u5408\u53f3\u624b\u6cd5\u5219\u3002\u7c7b\u4f3c\u4e8e\u529b\u8ddd\u7684\u8ba1\u7b97\u3002<\/p>\n

\u529b\u8ddd: \u5728\u7269\u7406\u5b66\u91cc\uff0c\u529b\u77e9\u662f\u4e00\u4e2a\u5411\u91cf\uff0c\u53ef\u4ee5\u88ab\u60f3\u8c61\u4e3a\u4e00\u4e2a\u65cb\u8f6c\u529b\u6216\u89d2\u529b\uff0c\u5bfc\u81f4\u51fa\u65cb\u8f6c\u8fd0\u52a8\u7684\u6539\u53d8\u3002\u50cf\u62e7\u87ba\u4e1d\uff0c\u4f7f\u87ba\u4e1d\u62e7\u7d27\u6216\u62e7\u5f00\u3002\u529b\u77e9\u7684\u5355\u4f4d\u662fN\u25cfm\u6216kN\u25cfm\uff0c\u7269\u7406\u5b66\u4e0a\u6307\u4f7f\u7269\u4f53\u8f6c\u52a8\u7684\u529b\u4e58\u4ee5\u5230\u8f6c\u8f74\u7684\u8ddd\u79bb\u3002<\/p>\n

 <\/p>\n

\u4e24\u4e2a\u5411\u91cf\u53c9\u4e58:
|crossProduct(A, B)| = |A| |B| sin<A,B>
crossProduct(A, B) = u |A| |B| sin<A,B>   \/\/ u: \u5355\u4f4d\u5411\u91cf\uff0c\u4f7f\u7528\u53f3\u624b\u6cd5\u5219\u53ef\u4ee5\u8ba1\u7b97\u5f97\u51fa<\/p>\n

\/\/ \u628aA, B\u6309x\/y\/z\u4e09\u4e2a\u65b9\u5411\u62c6\u5206\uff0c\u62c6\u5206\u6210\u4e09\u4e2a\u65b9\u5411\u540e\uff0c\u56e0\u4e3a\u4e92\u76f8\u5782\u76f4\uff0c\u540c\u65b9\u5411sin(0\u5ea6)=0\uff0c\u4e0d\u540c\u65b9\u5411sin(90\u5ea6)=1\u6216sin(-90\u5ea6)=-1\uff0c\u6240\u4ee5\u53ea\u6709\u4e0d\u540c\u65b9\u5411\u7684\u503c\u624d\u4f1a\u4fdd\u7559
\/\/ \u6839\u636e\u83b7\u53d6\u7ed3\u679c\u5bf9\u5e94\u7684\u6b63\u8d1f\u8f74\u7684\u4e0d\u540c\uff0c\u5f97\u51fa
\/\/ AxBy(\u6b63z\u8f74)  \u2013 AxBz(\u8d1fy\u8f74) 
\/\/ -AyBx(\u8d1fz\u8f74)  AyBz(\u6b63x\u8f74) 
\/\/ AzBx(\u6b63y\u8f74)   AzBy(\u8d1fx\u8f74) \u628a\u8fd9\u4e9b\u7ed3\u679c\u6574\u5408\u5c31\u53ef\u4ee5\u5f97\u5230<\/p>\n

crossProduct(A, B) = (AyBz \u2013 AzBy,    AzBx - AxBz,    AxBy - AyBx)<\/p>\n

\u884c\u5217\u5f0f\u8868\u8fbe i\uff0cj\uff0ck\u8868\u8fbe\u5355\u4f4dx\u8f74,y\u8f74,z\u8f74\u5411\u91cf
i        j       k
Ax  Ay   Az
Bx  By   Bz
crossProduct(A, B) = (AyBz \u2013 AzBy)i + (AzBx \u2013 AxBz)j + (AxBy \u2013 AyBx)k<\/p>\n

 <\/p>\n

(Owed by: \u6625\u591c\u559c\u96e8 http:\/\/blog.csdn.net\/chunyexiyu \u8f6c\u8f7d\u8bf7\u6807\u660e\u6765\u6e90)<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"\u5411\u91cf\u70b9\u79ef\u548c\u53c9\u4e58_\u5411\u91cf\u53c9\u4e58\u518d\u70b9\u4e58\u600e\u4e48\u8fd0\u7b97\u53c2\u8003\u6587\u6863\uff1a\u300a\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u300b\u8bbe\u5411\u91cf:V1(x1,y1,z1)V2(x2,y2,z2)\u5411\u91cf\u957f\u5ea6\uff1a(\u6807\u91cf)|V1|=\u6839...","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"_links":{"self":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts\/6261"}],"collection":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/comments?post=6261"}],"version-history":[{"count":0,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/posts\/6261\/revisions"}],"wp:attachment":[{"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/media?parent=6261"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/categories?post=6261"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mushiming.com\/wp-json\/wp\/v2\/tags?post=6261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}