Question

设向量OQ=(根号3,-1),向量OP=(cosa,sina),0

Answer 1

OQ=(√3,-1),|OQ| = 2,幅角 arg OQ = arc tan (-1/√3) = -π/6

(1) OP垂直OQ,所以,幅角相差 π/2,则 a= -π/6 + π/2 = π/3,所以,tan a = √3

(2) 由余弦定理 |PQ|^2 = |OQ|^2 + |OP|^2 - 2 |OQ|*|OP| cos (a+π/6) = 5 - 4 cos(a+π/6)

显然,最大值当且仅当 cos(a+π/6),即 a = 5π/6时,最大值为 |PQ|^2=9,|PQ|=3.