u=x1i+y1jandv=x2i+y2jau=ax1i+ay1jbv=bx2i+by2jau+bv=bx2i+by2j+ax1i+ay1jau+bv=(ax1+bx2)i+(ay1+by2)j∥au+bv∥=(ax1+bx2)2+(ay1+by2)2bu=bx1i+by1jav=ax2i+ay2jbu+av=(bx1+ax2)i+(by1+ay2)j∥bu+av∥=(bx1+ax2)2+(by1+ay2)2∥bu+av∥=∥au+bv∥(ax1+bx2)2+(ay1+by2)2=(bx1+ax2)2+(by1+ay2)2(ax1+bx2)2+(ay1+by2)2=(bx1+ax2)2+(by1+ay2)2(ax1+bx2)2−(bx1+ax2)2=(by1+ay2)2−(ay1+by2)2(ax1+bx2+bx1+ax2)(ax1+bx2−bx1−ax2)=(by1+ay2+ay1+by2)(by1+ay2−ay1−by2)((a+b)x1+(a+b)x2)((a−b)x1−(a−b)x2)=((a+b)y1+(a+b)y2)((b−a)y1−(b−a)y2)(a2−b2)(x12−x22)=−(a2−b2)(y12−y22)(x12−x22)=−(y12−y22)(x12−x22)=−y12+y22(x12+y12)=(x22+y22)(x12+y12)=(x22+y22)∴∥u1∥=∥v1∥
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