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匿名使用者 發問時間: 社會與文化語言 · 1 0 年前

幫解一題英文微積分

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Is the cross product associative? That is, let u, v and w be space vectors, u*v be the cross product of u and v. Does (u*v)*w generally equal to u*(v*w)? If you think it is, give reasons. If you think it is not, give a counterexample.

2 個解答

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  • 1 0 年前
    最佳解答

    Ok, answer is not.

    Let say u = [a1, a2, a3], v = [b1, b2, b3], and w = [c1, c2, c3]

    u x v = [a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1]

    v x w = [b2c3 - b3c2, b3c1 - b1c3, b1c2 - b2c1]

    The x-component for (u x v) x w is:

    (a3b1 - a1b3)*c3 - (a1b2 - a2b1)*c2

    The x-component for u x (v x w) is:

    a2*(b1c2 - b2c1) - a3*(b3c1 - b1c3)

    Obviously the above two don't equal.

    Similarly to show that y- and z-component of the result vectors don't equal to each other respectively.

    Therefore, cross product isn't associative.

  • 1 0 年前

    交叉產品是否是結合的? 即讓u, v和w是空間向量, u*v是u和v.交叉產品。 (u*v) *w是否一般合計對u* (v*w) ? 如果您認為它是,給原因。 如果您認為它不是,給反例。

    參考資料: 私の
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