向量a.(a+b)=a^2+a.b 【以下略去“向量”二字.】
又,|a+b|=√(a^2+2ab+b^2)
cos=a.(a+b)/|a||a+b|=cos60°.
(a^2+ab)=(1/2)|√2*√(a2+2ab+b^2).
2(2+ab)=√2(2+2ab+3)
(4+2ab)=√(10+4ab).兩邊平方：
(4+2ab)^2=10+4ab.
16+16ab+4(ab)^2=10+4ab
4(ab)^2+12ab+6=0.
2(ab)^2+6ab+3=0.
2(ab+3/2)^2-9/2+3=0.
2(ab+3/2)^2=3/2
(ab+3/2)=±√3/2.
ab=±√3/2-3/2.
ab=√3/2-3/2,或ab=-√3/2-3/2.
cos(a,b>=ab/|a||b|=(√3/2-3/2)/√2*√3=(1/2)(√3-3)/√6.
=(1/4)(√2-√6).
=arc[(1/4))(√2-√6)]
或,cos=-(1/4)(√2+√6).
=arc[-(1/4)(√2+√6)] " -" 表示為鈍角.