在平面直角坐标系中已知向量a={cos(α-20°),sin(α-20°)},向量b={cos(α+40°),sin(α+40°)}则|a-b| =
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/15 05:18:56
![在平面直角坐标系中已知向量a={cos(α-20°),sin(α-20°)},向量b={cos(α+40°),sin(α+40°)}则|a-b| =](/uploads/image/z/3600093-21-3.jpg?t=%E5%9C%A8%E5%B9%B3%E9%9D%A2%E7%9B%B4%E8%A7%92%E5%9D%90%E6%A0%87%E7%B3%BB%E4%B8%AD%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%3D%7Bcos%28%CE%B1-20%C2%B0%EF%BC%89%2Csin%EF%BC%88%CE%B1-20%C2%B0%EF%BC%89%7D%2C%E5%90%91%E9%87%8Fb%3D%EF%BD%9Bcos%EF%BC%88%CE%B1%2B40%C2%B0%EF%BC%89%2Csin%EF%BC%88%CE%B1%2B40%C2%B0%EF%BC%89%EF%BD%9D%E5%88%99%7Ca-b%7C+%3D)
在平面直角坐标系中已知向量a={cos(α-20°),sin(α-20°)},向量b={cos(α+40°),sin(α+40°)}则|a-b| =
在平面直角坐标系中已知向量a={cos(α-20°),sin(α-20°)},向量b={cos(α+40°),sin(α+40°)}则|a-b| =
在平面直角坐标系中已知向量a={cos(α-20°),sin(α-20°)},向量b={cos(α+40°),sin(α+40°)}则|a-b| =
|a-b|^2=[cos(α-20°)-cos(α+40°) ]^2 + [sin(α-20°)-sin(α+40°)]^2
=[cos(α-20°)]^2+[cos(α+40°)]^2-2cos(α-20°)*cos(α+40°)+[sin(α-20°)]^2+[sin(α+40°)]^2-2sin(α-20°)*sin(α+40°)
={[cos(α-20°)]^2+[sin(α-20°)]^2}+{[cos(α+40°)]^2+[sin(α+40°)]^2} -
2[cos(α-20°)*cos(α+40°)+sin(α-20°)*sin(α+40°]
=1+1-2cos[(α-20°)-(α+40°)]
=2-2cos(-60°)
=1
==>|a-b|=1
其中a^2表示a的平方
解:题中的两个向量可知是单位向量,则|a|=|b|=1;两向量的夹角为B=(a+40)-(a-20)=60;
而|a-b|即为两向量构成的三角形的第三边,由余弦定理得:
|a-b|^2=1^2+1^2-2X1X1Xcos60=1
则|a-b|=1