{"id":2124,"date":"2023-04-02T09:45:21","date_gmt":"2023-04-02T01:45:21","guid":{"rendered":"https:\/\/www.appblog.cn\/?p=2124"},"modified":"2023-04-05T20:42:13","modified_gmt":"2023-04-05T12:42:13","slug":"fundamentals-of-high-school-mathematics-plane-vectors","status":"publish","type":"post","link":"https:\/\/www.appblog.cn\/index.php\/2023\/04\/02\/fundamentals-of-high-school-mathematics-plane-vectors\/","title":{"rendered":"\u9ad8\u4e2d\u6570\u5b66\u57fa\u7840\uff1a\u5e73\u9762\u5411\u91cf"},"content":{"rendered":"

\u5e73\u9762\u5411\u91cf\u57fa\u672c\u6982\u5ff5<\/h2>\n

1\u3001\u5411\u91cf\uff1a\u65e2\u6709\u5927\u5c0f\uff0c\u53c8\u6709\u65b9\u5411\u7684\u91cf\u53eb\u505a\u5411\u91cf\uff0c\u5982\u529b\u3001\u4f4d\u79fb\u7b49
\n2\u3001\u6570\u91cf\uff1a\u53ea\u6709\u5927\u5c0f\uff0c\u6ca1\u6709\u65b9\u5411\u7684\u91cf\u79f0\u4e3a\u6570\u91cf\uff0c\u5982\u5e74\u9f84\u3001\u8eab\u9ad8\u3001\u957f\u5ea6\u3001\u9762\u79ef\u3001\u4f53\u79ef\u3001\u8d28\u91cf\u7b49<\/p>\n

\u96f6\u5411\u91cf\uff1a\u957f\u5ea6\u4e3a0\u7684\u5411\u91cf
\n\u5355\u4f4d\u5411\u91cf\uff1a\u957f\u5ea6\u4e3a1\u4e2a\u5355\u4f4d\u7684\u5411\u91cf<\/p>\n

<\/p>\n

\u6709\u5411\u7ebf\u6bb5\u8868\u793a\uff1a$\\overrightarrow{AB}$
\n\u6709\u5411\u7ebf\u6bb5\u957f\u5ea6\uff1a$|\\overrightarrow{AB}|$<\/p>\n

\u76f8\u7b49\u5411\u91cf\u53ca\u5e73\u884c\u5411\u91cf<\/h2>\n

\uff081\uff09\u76f8\u7b49\u5411\u91cf<\/p>\n

\u957f\u5ea6\u76f8\u7b49\u4e14\u65b9\u5411\u76f8\u540c\u7684\u5411\u91cf\u53eb\u505a\u76f8\u7b49\u5411\u91cf\u3002\u8bb0\u4e3a${\\bf {a}}={\\bf {b}}$\u3002<\/p>\n

\u4efb\u610f\u4e24\u4e2a\u76f8\u7b49\u7684\u975e\u96f6\u5411\u91cf\uff0c\u90fd\u53ef\u7528\u540c\u4e00\u6761\u6709\u5411\u7ebf\u6bb5\u6765\u8868\u793a\uff0c\u5e76\u4e14\u4e0e\u6709\u5411\u7ebf\u6bb5\u7684\u8d77\u70b9\u65e0\u5173\u3002\u5728\u5e73\u9762\u4e0a\uff0c\u4e24\u4e2a\u957f\u5ea6\u76f8\u7b49\u4e14\u65b9\u5411\u4e00\u81f4\u7684\u6709\u5411\u7ebf\u6bb5\u8868\u793a\u540c\u4e00\u4e2a\u5411\u91cf\u3002<\/p>\n

\uff082\uff09\u5e73\u884c\u5411\u91cf<\/p>\n

\u65b9\u5411\u76f8\u540c\u6216\u76f8\u53cd\u7684\u975e\u96f6\u5411\u91cf\u53eb\u505a\u5e73\u884c\u5411\u91cf\u3002\u8bb0\u4e3a${\\bf {a}}\/\/{\\bf {b}}$\u3002<\/p>\n

\u89c4\u5b9a\u96f6\u5411\u91cf\u4e0e\u4efb\u4f55\u5411\u91cf\u90fd\u5e73\u884c\u3002<\/p>\n

\u4efb\u4e00\u7956\u5e73\u884c\u5411\u91cf\u90fd\u53ef\u4ee5\u5e73\u79fb\u5230\u540c\u4e00\u76f4\u7ebf\u4e0a\uff0c\u56e0\u6b64\u5e73\u884c\u5411\u91cf\u4e5f\u53eb\u5171\u7ebf\u5411\u91cf\u3002<\/p>\n

\u5411\u91cf\u8fd0\u7b97<\/h2>\n

\u5411\u91cf\u52a0\u6cd5<\/h3>\n

\u8bbe${\\bf {a}}=\\overrightarrow{AB}$\uff0c${\\bf {b}}=\\overrightarrow{BC}$\uff0c\u5219<\/p>\n

${\\bf {a}} + {\\bf {b}} = \\overrightarrow{AC}$<\/p>\n

\uff081\uff09${\\bf {a}} + {\\bf {b}} = {\\bf {b}} + {\\bf {a}}$
\n\uff082\uff09$|{\\bf {a}} + {\\bf {b}}| \u2264 |{\\bf {a}}| + |{\\bf {b}}| $<\/p>\n

\u5411\u91cf\u51cf\u6cd5<\/h3>\n

\u8bbe${\\bf {a}}=\\overrightarrow{OA}$\uff0c${\\bf {b}}=\\overrightarrow{OB}$\uff0c\u5219<\/p>\n

${\\bf {a}} – {\\bf {b}} = \\overrightarrow{BA}$<\/p>\n

\u53ea\u9700\u8bb0\u4f4f\uff1a\u8fde\u63a5\u4e24\u5411\u91cf\u7684\u7ec8\u70b9\uff0c\u7bad\u5934\u6307\u5411\u88ab\u51cf\u5411\u91cf<\/p>\n

\uff081\uff09${\\bf {a}} – {\\bf {b}} = – {\\bf {b}} + {\\bf {a}}$
\n\uff082\uff09$|{\\bf {a}} – {\\bf {b}}| \u2264 |{\\bf {a}}| + |{\\bf {b}}| $<\/p>\n

\u5411\u91cf\u4e58\u6cd5<\/h3>\n

\u8bbe\u5b9e\u6570$\\lambda$\u548c\u5411\u91cf${\\bf {a}}$\uff0c\u5219\u5411\u91cf\u7684\u6570\u4e58\u8bb0\u4f5c$\\lambda{\\bf {a}}$<\/p>\n

\uff081\uff09\u957f\u5ea6$|\\lambda{\\bf {a}}| = |\\lambda||{\\bf {a}}|$
\n\uff082\uff09\u5bf9\u4efb\u610f\u975e\u96f6\u5411\u91cf${\\bf {a}}$\uff0c\u5411\u91cf$\\frac{\\bf {a}}{|{\\bf {a}}|}$\u662f\u4e0e\u5411\u91cf${\\bf {a}}$\u540c\u5411\u7684\u5355\u4f4d\u5411\u91cf
\n\uff083\uff09$\\lambda{\\bf {a}}$\u7684\u51e0\u4f55\u610f\u4e49\u5c31\u662f\u5c06\u5411\u91cf${\\bf {a}}$\u6cbf\u7740${\\bf {a}}$\u7684\u65b9\u5411\u6216\u53cd\u65b9\u5411\u6269\u5927\u6216\u7f29\u5c0f$|\\lambda|$\u500d<\/p>\n

\u5411\u91cf\u6570\u4e58\u7684\u8fd0\u7b97\u5f8b<\/h3>\n

\u8bbe$\\lambda$\u3001$\\mu$\u4e3a\u5b9e\u6570\uff0c\u5219<\/p>\n

\uff081\uff09$\\lambda(\\mu{\\bf {a}}) = (\\lambda\\mu){\\bf {a}}$
\n\uff082\uff09$(\\lambda+\\mu){\\bf {a}} = \\lambda{\\bf {a}} + \\mu{\\bf {a}}$
\n\uff083\uff09$(\\lambda-\\mu){\\bf {a}} = \\lambda{\\bf {a}} – \\mu{\\bf {a}}$
\n\uff084\uff09$\\lambda({\\bf {a}} + {\\bf {b}}) = \\lambda{\\bf {a}} + \\lambda{\\bf {b}}$
\n\uff085\uff09$\\lambda({\\bf {a}} – {\\bf {b}}) = \\lambda{\\bf {a}} – \\lambda{\\bf {b}}$
\n\uff086\uff09$(-\\lambda){\\bf {a}} = -(\\lambda{\\bf {a}}) = \\lambda(-{\\bf {a}})$<\/p>\n

\u5411\u91cf\u7684\u7ebf\u6027\u8fd0\u7b97<\/h3>\n

\u5411\u91cf\u7684\u52a0\u3001\u51cf\u3001\u6570\u4e58\u8fd0\u7b97\u7edf\u79f0\u4e3a\u5411\u91cf\u7684\u7ebf\u6027\u8fd0\u7b97\uff0c\u5bf9\u4e8e\u4efb\u610f\u5411\u91cf${\\bf {a}}$\uff0c${\\bf {b}}$\u4ee5\u53ca\u4efb\u610f\u5b9e\u6570$\\lambda$\uff0c$\\mu_1$\uff0c$\\mu_2$\uff0c\u6052\u6709<\/p>\n

$\\lambda(\\mu_1{\\bf {a}}+\\mu_2{\\bf {b}}) = \\lambda\\mu_1{\\bf {a}} + \\lambda\\mu_2{\\bf {b}}$<\/p>\n

\u4e24\u4e2a\u5411\u91cf\u6570\u91cf\u79ef\u7684\u5750\u6807\u8868\u793a<\/h3>\n

\u6570\u91cf\u79ef\u7684\u5750\u6807\u5f62\u5f0f\uff1a\u8bbe${\\bf {a}}=(x_1, y_1)$\uff0c${\\bf {b}}=(x_2, y_2)$\uff0c\u5219${\\bf {a}}\u00b7{\\bf {b}}=x_1x_2+y_1y_2$<\/p>\n

\u6570\u91cf\u79ef\u5750\u6807\u5f62\u5f0f\u7684\u63a8\u5bfc\uff1a\u53d6\u4e0e$x$\u8f74\u3001$y$\u8f74\u5206\u522b\u540c\u5411\u7684\u4e24\u4e2a\u5355\u4f4d\u5411\u91cf${\\bf {i}}$\u3001${\\bf {j}}$\uff0c\u5219${\\bf {a}}=(x_1, y_1)=x_1{\\bf {i}}+y_1{\\bf {j}}$\uff0c${\\bf {b}}=(x_2, y_2)=x_2{\\bf {i}}+y_2{\\bf {j}}$\u3002\u7531\u6570\u91cf\u79ef\u7684\u5b9a\u4e49\u53ef\u77e5\uff1a${\\bf {i}}\u00b7{\\bf {i}}=1$\uff0c${\\bf {j}}\u00b7{\\bf {j}}=1$\uff0c${\\bf {i}}\u00b7{\\bf {j}}=0$\uff0c${\\bf {j}}\u00b7{\\bf {i}}=0$\uff0c\u6240\u4ee5<\/p>\n

${\\bf {a}}\u00b7{\\bf {b}}=(x_1{\\bf {i}}+y_1{\\bf {j}})(x_2{\\bf {i}}+y_2{\\bf {j}})=x_1x_2{\\bf {i}}\u00b7{\\bf {i}}+x_1y_2{\\bf {i}}\u00b7{\\bf {j}}+x_2y_1{\\bf {j}}\u00b7{\\bf {i}}+y_1y_2{\\bf {j}}\u00b7{\\bf {j}}=x_1x_2+y_1y_2$<\/p>\n

\u5373\u4e24\u4e2a\u5411\u91cf\u7684\u6570\u91cf\u79ef\u7b49\u4e8e\u5b83\u4eec\u5bf9\u5e94\u5750\u6807\u7684\u4e58\u79ef\u4e4b\u548c\u3002<\/p>\n

\u5411\u91cf\u7684\u6a21\u4e0e\u5782\u76f4\u5173\u7cfb\u7684\u5750\u6807\u8868\u793a<\/h3>\n

\u8bbe${\\bf {a}}=(x, y)$\uff0c\u6709$|{\\bf {a}}|=\\sqrt{x^2+y^2}$<\/p>\n

\u5411\u91cf\u5782\u76f4\u7684\u5750\u6807\u8868\u793a\uff1a\u7531\u5411\u91cf\u6570\u91cf\u79ef\u7684\u5b9a\u4e49\uff0c${\\bf {a}}\u00b7{\\bf {b}}=|{\\bf {a}}||{\\bf {b}}|\\cos \\theta\u00b7=x_1x_2+y_1y_2$<\/p>\n

${\\bf {a}} \\bot {\\bf {b}} \\Longleftrightarrow {\\bf {a}}\u00b7{\\bf {b}} = 0(|{\\bf {a}}|\u00b7|{\\bf {b}}|\u22600) \\Longleftrightarrow x_1x_2+y_1y_2 = 0$<\/p>\n

\u5411\u91cf\u5939\u89d2\u7684\u5750\u6807\u8868\u793a<\/h3>\n

\u8bbe${\\bf {a}}=(x_1, y_1)$\uff0c${\\bf {b}}=(x_2, y_2)$\uff0c$\\theta$\u662f${\\bf {a}}$\u4e0e${\\bf {b}}$\u7684\u5939\u89d2\uff0c\u7531\u6570\u91cf\u79ef\u7684\u5b9a\u4e49<\/p>\n

$${\\bf {a}}\u00b7{\\bf {b}} = |{\\bf {a}}|\u00b7|{\\bf {b}}|\\cos \\theta$$<\/p>\n

$$\\cos \\theta = \\frac{ {\\bf {a}}\u00b7{\\bf {b}} }{|{\\bf {a}}|\u00b7|{\\bf {b}}|} = \\frac{x_1x_2+y_1y_2}{\\sqrt{x_1^2+y_1^2}\\sqrt{x_2^2+y_2^2}}$$<\/p>\n

\u5229\u7528\u6b64\u516c\u5f0f\uff0c\u53ef\u76f4\u63a5\u6c42\u51fa\u4e24\u5411\u91cf\u7684\u5939\u89d2<\/p>\n

\u8bc1\u660e\uff1a<\/p>\n

\u8bbe${\\bf {c}} = {\\bf {a}} – {\\bf {b}}$\uff0c\u5219<\/p>\n

${\\bf {c}}^2 = ({\\bf {a}} – {\\bf {b}})^2$<\/p>\n

$|{\\bf {c}}|^2 = |{\\bf {a}}|^2 + |{\\bf {b}}|^2 – 2{\\bf {a}}\u00b7{\\bf {b}}$<\/p>\n

\u53c8 $|{\\bf {c}}|^2 = |{\\bf {a}}|^2 + |{\\bf {b}}|^2 – 2|{\\bf {a}}||{\\bf {b}}|\\cos \\theta$ (\u4e09\u89d2\u4f59\u5f26\u5b9a\u7406)<\/p>\n

\u5373\uff1a${\\bf {a}}\u00b7{\\bf {b}} = |{\\bf {a}}||{\\bf {b}}|\\cos \\theta$<\/p>\n

\u4e09\u89d2\u4f59\u5f26\u5b9a\u7406<\/h3>\n

$$
\nc^2 = a^2 + b^2 – 2ab \\cos \u2220ACB = a^2 + b^2 – 2ab \\cos \\theta
\n$$<\/p>\n

\u7279\u522b\u5730\uff0c\u5f53$\u2220ACB=90^\u3002$\uff0c\u5373\u52fe\u80a1\u5b9a\u7406<\/p>\n

\u8bc1\u660e\uff1a<\/p>\n

$c = b \\cos \\alpha + a \\cos \\beta \\Longleftrightarrow c^2 = bc \\cos \\alpha + ac \\cos \\beta$\uff0c\u540c\u7406<\/p>\n

$
\n\\left \\{
\n\\begin{array} {c}
\na^2 = ab \\cos \\gamma + ac \\cos \\beta \\
\nb^2 = ab \\cos \\gamma + bc \\cos \\alpha
\n\\end{array}
\n\\right.
\n$<\/p>\n

\u5373\uff1a$a^2 + b^2 = 2ab \\cos \\gamma + ac \\cos \\beta + bc \\cos \\alpha = c^2 + 2ab \\cos \\gamma$<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5e73\u9762\u5411\u91cf\u57fa\u672c\u6982\u5ff5 1\u3001\u5411\u91cf\uff1a\u65e2\u6709\u5927\u5c0f\uff0c\u53c8\u6709\u65b9\u5411\u7684\u91cf\u53eb\u505a\u5411\u91cf\uff0c\u5982\u529b\u3001\u4f4d\u79fb\u7b49 2\u3001\u6570\u91cf\uff1a\u53ea\u6709\u5927\u5c0f\uff0c\u6ca1\u6709\u65b9\u5411\u7684\u91cf\u79f0\u4e3a […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[509],"tags":[],"class_list":["post-2124","post","type-post","status-publish","format-standard","hentry","category-mathematics-fundamentals"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2124","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2124"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2124\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2124"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2124"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2124"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}