標題:
數學 vector
發問:
1. p,q and r are non zero vectors. If ap x q + bq x r + cr x p =0, where a,b and c are non zero constants, show that p,q and r are coplanar. 2. Suppose a,b and c are not coplanar. Prove that axb ,bxc and cxa are not coplanar.
1. It is obvious that If p//q or q//r or p//r, then p, q and r are coplanar Suppose that none of p, q, r are parallel. Without loss of generality, let a≠0. (a pxq + b qxr + c rxp)?r = 0?r = 0 a (pxq)?r + 0 + 0 =0 so that (pxq)?r =0, hence p, q and r are coplanar. 2. Suppose m(axb)+n(bxc)+k(cxa) = 0 (we should prove that m=n=k.) If m≠0, then [m(axb)+n(bxc)+k(cxa)]?c = 0?c =0 m(axb)?c + 0 + 0 = 0, so m=0 (since (axb)?c ≠ 0) Similarly, n=k=0, thus (axb), (bxc), (cxa) are linearly indep. hence, they are not coplanar.
其他解答:
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數學 vector
發問:
1. p,q and r are non zero vectors. If ap x q + bq x r + cr x p =0, where a,b and c are non zero constants, show that p,q and r are coplanar. 2. Suppose a,b and c are not coplanar. Prove that axb ,bxc and cxa are not coplanar.
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最佳解答:1. It is obvious that If p//q or q//r or p//r, then p, q and r are coplanar Suppose that none of p, q, r are parallel. Without loss of generality, let a≠0. (a pxq + b qxr + c rxp)?r = 0?r = 0 a (pxq)?r + 0 + 0 =0 so that (pxq)?r =0, hence p, q and r are coplanar. 2. Suppose m(axb)+n(bxc)+k(cxa) = 0 (we should prove that m=n=k.) If m≠0, then [m(axb)+n(bxc)+k(cxa)]?c = 0?c =0 m(axb)?c + 0 + 0 = 0, so m=0 (since (axb)?c ≠ 0) Similarly, n=k=0, thus (axb), (bxc), (cxa) are linearly indep. hence, they are not coplanar.
其他解答:
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