一平面与空间四边形[tex=3.143x1.286]REaUoNxha/GBn3DE8cgfDA==[/tex]的边[tex=1.571x1.286]cHJ4KDAad01mWuGaiQQpfA==[/tex]，[tex=1.571x1.286]hOo99m7YJCAnVf2cQGX8dQ==[/tex]，[tex=1.643x1.286]lIB/SPc41Ri5ohE6MtARRw==[/tex]，[tex=1.571x1.286]Mr2N+LwPSspF/qoGlNiX3w==[/tex]分别交于[tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex]，[tex=0.786x1.286]gvyykdQdNBydRqWi9I4iuA==[/tex]，[tex=0.786x1.286]yokTf2U2Z7kNGUXMm22GjQ==[/tex]，[tex=0.714x1.286]yQZEV57S9rHjYvgfJydTyg==[/tex]，则[tex=10.929x2.214]1kNiW/vR7aPwuPclPdyQTrd5f2hUN6sa/AYPAexOWY2EeKKDCn4ALPRSBlElrAsCEaSB4g+xJV5Gj4wQotf46J62GeFoDi4YgRnrpTFVvVZaCPwA2dKf8xGfsaDRGpm/[/tex]。试证之。[img=746x362]177ed3d5a576d74.png[/img]

2022-06-05

一平面与空间四边形[tex=3.143x1.286]REaUoNxha/GBn3DE8cgfDA==[/tex]的边[tex=1.571x1.286]cHJ4KDAad01mWuGaiQQpfA==[/tex]，[tex=1.571x1.286]hOo99m7YJCAnVf2cQGX8dQ==[/tex]，[tex=1.643x1.286]lIB/SPc41Ri5ohE6MtARRw==[/tex]，[tex=1.571x1.286]Mr2N+LwPSspF/qoGlNiX3w==[/tex]分别交于[tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex]，[tex=0.786x1.286]gvyykdQdNBydRqWi9I4iuA==[/tex]，[tex=0.786x1.286]yokTf2U2Z7kNGUXMm22GjQ==[/tex]，[tex=0.714x1.286]yQZEV57S9rHjYvgfJydTyg==[/tex]，则[tex=10.929x2.214]1kNiW/vR7aPwuPclPdyQTrd5f2hUN6sa/AYPAexOWY2EeKKDCn4ALPRSBlElrAsCEaSB4g+xJV5Gj4wQotf46J62GeFoDi4YgRnrpTFVvVZaCPwA2dKf8xGfsaDRGpm/[/tex]。试证之。[img=746x362]177ed3d5a576d74.png[/img]

答案：

证明：如图，我们以[tex=0.786x1.286]q1djlrfSWHAqH21hBgtrSw==[/tex]为原点，以[tex=3.857x1.857]5kxdtulOg1rJWKnrGXaX+xmqMVnqMxUXbXqhKsz9r4N/MaICYvbe/muxiG9bAqqj5WTkvV4xblWojJ0+CG6gOg==[/tex]，[tex=3.857x1.857]5kxdtulOg1rJWKnrGXaX+/GnP2toQYFNBjzMOJfEc2ckQnpnjRbbpkoRdiu4oVJDQbJa20ZaD9Kl/lxwA+ulsQ==[/tex]，[tex=3.929x1.857]5kxdtulOg1rJWKnrGXaX+54qmMZeZlYCNFG9btaH/qjHeGK7Va8vg3z7TW73GcEaFJJcyXR7scpWWg5nfz3C0A==[/tex]为基向量构作一个仿射坐标系[tex=7.286x1.857]4f1dGatS9sesc01k/ehnqYg7VRrRODrNapB9iZpLVRGNe3mVTUuFUKcargQEy0El7HwB6JVl1ULqNyPB6QQ7yub3l2tO6CfQZfPBApKLHFU=[/tex]。令[tex=4.929x1.857]5kxdtulOg1rJWKnrGXaX+6BHpVvtxL5Qqt/tMYBLX349muyZd1fM/ezHzydEzRp+GsFlxvntqKYBeMJJedN8Zg==[/tex]，[tex=5.214x1.857]5kxdtulOg1rJWKnrGXaX+1gxajqn0MHXyn5zOBw3/dQmsojlrudKpymeF9+XB8mH6JtkMd5I6Jrt1USY+VyaRg==[/tex]，[tex=4.643x1.857]5kxdtulOg1rJWKnrGXaX+y9e+5SptTBhE7iY4tkcAGKm8Bk77eNTjloJrVdF06MdMNDgL1Caw7TXTTE1M3lZqg==[/tex]，[tex=4.857x1.857]5kxdtulOg1rJWKnrGXaX+9h8snD9KMzIii9UFVlpTh56kyr2UbxFXMNnC9nvhsIPzI+DDA1DrlJPcT885Mc3yA==[/tex]。设[tex=14.643x1.857]5kxdtulOg1rJWKnrGXaX+yLEBlxPlrdpogcXRXWa/VtGds5hMqmk57uwwj51wCGvqv064LQ+TECBE0StNjjrP6ImyozvmOPC2MpAMetJMDM9EIPR8hjHQeWegfKJN8kPnIzj5iI7MRYQVK8AvdoRZw==[/tex]，[tex=14.357x1.857]5kxdtulOg1rJWKnrGXaX+x1KUngF+ILUXPvhDrBg+znSbBgv/eWVOp0aSUe3dIfIgqVz4UBsoySZZ/d6YhaUMG/NG2eMIelJdTbHUFx56PMOdOqkkGwJ7f9T3MsrXKkGKoYwrhNZ26x3KlmIRN3KTggiD7YQ+1HuA7TaVfPkAoU=[/tex]，则[tex=16.429x1.857]5kxdtulOg1rJWKnrGXaX+1gxajqn0MHXyn5zOBw3/dQsQEhh1Hm8uWW9YEQHb9zzIGHyjSI9gICEJWhuxRSelhXDs/+BdMQhqIa/2loKgm4d6EsXry7Qq7L/KtTCMIGSMGl1ULohQxn3PxnG0qpFjPEmdNO921TqaebdIMDG0Rs=[/tex]，[tex=15.643x1.857]5kxdtulOg1rJWKnrGXaX+7YFWNyhv+ztclYvz4DFLVSkJIuMHJZ6gBnB3HgJcMRcHFVGUIAp58E20xrXVPWlRy5lFH6BpKMZJw4mNDikmjvWUiRcp2hIBdchQNtOCaP3ZnggjFxm3y48pL4ugH9GIFBcw3306YVmkco55/t+Ync=[/tex]。由[tex=4.857x1.857]5kxdtulOg1rJWKnrGXaX+9GZ3v4OumZze8IfIVERBCheKy7s2katKnpWexaufmAZfWRd+EJHmkFPK3BNIIbQ9A==[/tex]可得：[tex=8.857x2.929]GE56u9QCDTqcLxZ66HADylc7Ytd9GwJNLNI0ii2e6PebybZLVfJgh9Dbn8yI/ENZKwFADfj1neAwKM02lXLHPOjFTpiXNY56KtWn45lqGQM=[/tex]，即[tex=7.786x5.357]GE56u9QCDTqcLxZ66HADytiLVf1os2abmcOwMqrWz5aLkzhWdAN3kGa9LsyPXJFNRdpaEh01LZb1++C5ekn4hNbFxu1WQIs6nWn9wJIzweFK4lzlvP3X/8KZuLS1qHGa[/tex]。又因[tex=15.286x1.857]5kxdtulOg1rJWKnrGXaX+8KVLTPFZq1Z7qxq8GJO2N3gE/7YVDcqiZK7j/umd4T/uRt+8+lOPx0duqyDZNVrLMWnhRn2BaOE3zkFfAUlZF+PROx/0yKVkWbNdUBYZXcyVOKuPTwSOU5DnWR/x5WI0TrUbwsonglaVHvlIcOzsT8=[/tex]，[tex=16.786x1.857]5kxdtulOg1rJWKnrGXaX+8WVRZDXJXHuvURlZ46X9CEiynWkWSiFk8LmRRAK6pilniu2dBVc6BaqcbSIsGwuV7YBoSgN/TSLJyjq6z2tRW8Ca/se4jt+6IYDoArfD/DwG/6gLH6DXz5L/RaNYXMh7uSeQOxMttq8FQzsFEiCHcg=[/tex]。由[tex=4.643x1.857]5kxdtulOg1rJWKnrGXaX+y9e+5SptTBhE7iY4tkcAGKm8Bk77eNTjloJrVdF06MdMNDgL1Caw7TXTTE1M3lZqg==[/tex]可得：[tex=9.571x2.929]GE56u9QCDTqcLxZ66HADyjeJs2zJ8jzwAs4zJnlxB9oAiPcfQNwvuKGprikbmzmFvBSUvlIittgC6I4E0vYL+7K3LDjjJeTkf2NCiIgDBuk=[/tex]即[tex=8.429x5.214]GE56u9QCDTqcLxZ66HADyoscsEC822nJ0fwP1KEIc09ZLJkIup8+d7G1yz9CLkBY23O7h/H6QxqGRJ4HfrC+ck8vWBAmAC9SvukS/bZRm9oU8YEk/tQJ8K4roTyCWe10[/tex]。所以[tex=18.857x2.286]1kNiW/vR7aPwuPclPdyQTrd5f2hUN6sa/AYPAexOWY2EeKKDCn4ALPRSBlElrAsCEaSB4g+xJV5Gj4wQotf46J62GeFoDi4YgRnrpTFVvVbT60Lf8pLb4pWW/u/uvNFjxmY9M5WiBZZJtkLNyZzsbVDn3mxO7hBuhxpTQ60ysh4sM00d0scP6tArLVMWTv/J[/tex]。(*)又，[tex=14.786x1.857]5kxdtulOg1rJWKnrGXaX+y8HWMGzcsBkDniNZZQVd5aBYSl/kG2HH3wupxd9hBephxw0ePWP/b+dqGecssy0BrXTpPsVjCOgKL6sz493NRG6nav1omJViPj+B9ICmudBoW4hNCozMPPYcJCTwfbxfoq5v1M2cR0NrGEA2u0s4t8=[/tex]，[tex=11.929x1.857]5kxdtulOg1rJWKnrGXaX+/zdyQaHnixjUMeWdQecj/sXJLZPL7sklAmRkL2KBvp0GLQbCGUOFT0hpEJTPTekK60y9Z3olKvqofMhkm/zrdUMNCxkOeYpXY9SYgAr4nI5pVXxuE1XCrNC11I/VbjZCQz836+zT/KfJEwzrLZlbKA=[/tex]，[tex=15.071x1.857]5kxdtulOg1rJWKnrGXaX+xu0uw+186uLEp0muvhJLJ2fIHkn3zsFNaHSpJQPnNJ1GqONtfomPATPKSUJED3QAgNJc3NWNXMjvLIH557jlTRJF1kbFi1JRZM0ZAF/bR+MQUe1VehaPb+PR5y12ypQuPq7Y6VjSaLT2MIghJhX8jQsdf4mlEx7whm+7XtnBk7xcXa9pB3rh5MKSZggEerueQ==[/tex]，因为这3个向量共面，所以[tex=19.429x3.643]1xLK2S2fjz/DkWdie5OKhRtnyCR+h4B2niWLADHgkyFK0eTZROXaQ+Ommd/bWS63mrf2a7eQKv/i+3YeCSwTHoTz1bvtpPr74Ds1XE8zB9QBxDiY7uo1aGKdDS17R6zLMeaf6TW6Z3FZG2BgSz6S9A==[/tex]。从而[tex=24.786x1.286]JmIdnAiWoQ9cXWYN+Y3gtWHzgeSxBNrWr2a44eiGeuCe5NQ7U6rnvxOIqSiJMTjq[/tex]。将上式代入(*）即得：[tex=10.929x2.214]1kNiW/vR7aPwuPclPdyQTrd5f2hUN6sa/AYPAexOWY2EeKKDCn4ALPRSBlElrAsCEaSB4g+xJV5Gj4wQotf46J62GeFoDi4YgRnrpTFVvVZaCPwA2dKf8xGfsaDRGpm/[/tex]。

举一反三

内容

0
两个正方形[tex=3.143x1.286]REaUoNxha/GBn3DE8cgfDA==[/tex]、[tex=3.143x1.286]xDRdx4umKeSEbPtNV3E/fQ==[/tex]的边长都是[tex=0.571x1.286]mRKL/orzOudCEARA8qn3Kw==[/tex]，其中[tex=0.786x1.286]/aLPP1sXG9WQPxIsGVtWrg==[/tex]为正方形[tex=3.143x1.286]REaUoNxha/GBn3DE8cgfDA==[/tex]的中心，[tex=1.571x1.286]8Fzac3+ZrjQZLUQj0V/ubg==[/tex]、[tex=1.571x1.286]QESKVt2E33NLJbUfExf2mQ==[/tex]分别交[tex=1.643x1.286]lIB/SPc41Ri5ohE6MtARRw==[/tex]、[tex=1.571x1.286]hOo99m7YJCAnVf2cQGX8dQ==[/tex]于[tex=0.929x1.286]+6R6Ey5borUsIf6RDxJ0vA==[/tex]、[tex=0.929x1.286]nrJzN9qRndstwtgYfof7gw==[/tex]，求四边形[tex=3.357x1.286]gNH3AR2tj5aj/Joh+/CK4Q==[/tex]的面积。
1
6个顶点11条边的所有非同构的连通的简单非平面图有[tex=2.143x2.429]iP+B62/T05A6ZTM0eeaWiQ==[/tex]个，其中有[tex=2.143x2.429]ndZSw3zT0QTOVLVdoUto1Q==[/tex]个含子图[tex=1.786x1.286]J+vVZa2YaMpc6mJBbqVvWw==[/tex]，有[tex=2.143x2.429]lmhx48evnQMhi03NovPXig==[/tex]个含与[tex=1.214x1.214]kFXZ1uR8GjycbJx+Ts2kyQ==[/tex]同胚的子图。供选择的答案[tex=3.071x1.214]3KinXFh3SXhZ7nIe1y9KEV6aadxhhJWeEy6Dij1iObdMUZkY6ZA5J2dVVjPSuhEf[/tex]：(1) 1 ;(2) 2 ;(3) 3 ; (4) 4 ;(5) 5 ;(6) 6 ; (7) 7 ; (8) 8 。
2
求函数[tex=3.286x1.429]kdT+eIE7CHPynuN6CaN40g==[/tex]（抛物线）隐函数的导数[tex=1.071x1.429]BUw1BPFU3fsJlAl/vt9M9w==[/tex]当x=2与y=4及当x=2与y=0时,[tex=0.786x1.357]Hq6bf3CacUy07X+VImUMaA==[/tex]等于什么?
3
以随机变量 [tex=0.857x1.0]N7iCrOsS+NNEUUlnsYCi1g==[/tex] 表示某游乐园内一主题商店从早晨开园起直到第一个游客到达的等待时间(单位：分钟 ), [tex=0.857x1.0]N7iCrOsS+NNEUUlnsYCi1g==[/tex] 的分布函数为[tex=12.643x3.357]+rmdHPH4CZj7YVOHS1cgeEZgjq4yS+iXNUsb/lBzTzhXnvymBT0ZLOmMiLd8nFXnkgGSIA2+deg26wTIu3uRnIm2M9uDO8JyL/yc9vazoP54Sdh8wWgNczOX6Kfzy+xjnlwJAhn2nTeBt86WzxbuFQ==[/tex]，求 1) [tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex]( 等待时间至多 3 分钟 ) ;(2) [tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex](等待时间至少 4 分钟 ) ;(3) [tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex]( 等待时间 3 分钟到 4 分钟 ) ;(4) [tex=0.786x1.286]dSWbQCTjdbLxKy7q0ps2gg==[/tex](等待时间恰好 2.5 分钟 ) ;(5) [tex=0.857x1.0]N7iCrOsS+NNEUUlnsYCi1g==[/tex] 的概率密度函数.
4
设[tex=1.571x1.286]fyE3LBxTKh2vAJHvxikdUA==[/tex]的直径[tex=1.5x1.286]1I/urUM/I6J252uekLAoyw==[/tex]垂直线段[tex=1.571x1.286]hOo99m7YJCAnVf2cQGX8dQ==[/tex]于[tex=0.857x1.286]s+r8LBAs3scxfl88DGExcg==[/tex]，连接[tex=1.571x1.286]aR1a8Eu3rZLX3flcxLOVFw==[/tex]、[tex=1.571x1.286]isotx5z8tmSlkvJ01wOD4g==[/tex]交圆于[tex=0.786x1.286]YggwMQ4w3PxfhkmL0NfgdQ==[/tex]、[tex=0.786x1.286]BlkXDnmzWHxe4M6E9LlofQ==[/tex]，则[tex=0.786x1.286]q1djlrfSWHAqH21hBgtrSw==[/tex]、[tex=0.786x1.286]YggwMQ4w3PxfhkmL0NfgdQ==[/tex]、[tex=0.786x1.286]BlkXDnmzWHxe4M6E9LlofQ==[/tex]、[tex=0.786x1.286]TKU5UzNEMzEJwORo6mbEYA==[/tex]共圆。