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        "quote": "> [!example] <a class=\"object-permalink\" href=\"#example-5-1\">例 5.1</a> \n>\n> 普通三维向量 $a$、$b$ 等的空间是一个欧氏向量空间，通常记作 ${ \\mathbb E } ^ { 3 }$，其相对于通常的标量积。\n>\n> $$\n> \\mathbf {a} \\cdot \\mathbf {b} = a _ {1} b _ {1} + a _ {2} b _ {2} + a _ {3} b _ {3} = | \\mathbf {a} | | \\mathbf {b} | \\cos \\theta\n> $$\n>\n> 其中 $| \\mathbf { a } |$ 是向量 $a$ 的长度或量值，$\\theta$ 是 $a$ 与 $b$ 之间的夹角。条件 (RIP1) 和 (RIP2) 易于验证，而 (RIP3') 则可由\n>\n> $$\n> \\mathbf {a} \\cdot \\mathbf {a} = \\mathbf {a} ^ {2} = (a _ {1}) ^ {2} + (a _ {2}) ^ {2} + (a _ {3}) ^ {2} > 0 \\quad \\text { if } \\quad \\mathbf {a} \\neq \\mathbf {0}.\n> $$\n>\n> 这推广为 $\\mathbb { R } ^ { n }$ 上的一个正定内积，\n>\n> $$\n> \\mathbf {a} \\cdot \\mathbf {b} = a _ {1} b _ {1} + a _ {2} b _ {2} + \\dots + a _ {n} b _ {n} = \\sum_ {i = 1} ^ {n} a _ {i} b _ {i},\n> $$\n>\n> 所得的欧氏向量空间记为 $\\mathbb { E } ^ { n }$。\n>\n向量 $w$ 的 <u id=\"concept-5-magnitude\" class=\"concept-term\" data-concept-identity=\"urn:uuid:dab77b33-384c-47d5-b9d2-1649b126f195\" data-concept-route=\"/concept-identities#concept-dab77b33-384c-47d5-b9d2-1649b126f195\">量值(magnitude)</u> 定义为 $w \\cdot w$。注意，在伪欧氏空间中，非零向量的量值可能为负或为零，但在欧氏空间中它始终是一个正量。欧氏空间中向量的 <u id=\"concept-5-length\" class=\"concept-term\" data-concept-identity=\"urn:uuid:e98e09ca-6bd2-49f0-92f7-1860bb397f02\" data-concept-route=\"/concept-identities#concept-e98e09ca-6bd2-49f0-92f7-1860bb397f02\">长度(length)</u> 定义为量值的平方根。",
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