y=ρsinθ=4sin 7π/6=-2.

故点M的直角坐标为(-2√3,-2,1).

答案(-2√3,-2,1)

5若点M的柱坐标为(7"," π/2 "," 7),点M的球坐标为(r,φ,θ),则r=　　　　.

解析∵(ρ,θ,z)=(7"," π/2 "," 7),

设点M的直角坐标为(x,y,z),

则x2+y2=ρ2=49,

∴r=√(x^2+y^2+z^2 )=√(49+7^2 )=7√2.

答案7√2

6已知空间点P的柱坐标为(6"," π/3 "," 4),则点P关于z轴的对称点的柱坐标为　　　　.

答案(6"," 4π/3 "," 4)

7把下列用柱坐标表示的点用直角坐标表示出来.

(1)(2,0,-2);(2)(π,π,π).

解设点的直角坐标为(x,y,z).

(1)∵(ρ,θ,z)=(2,0,-2),∴{■(x=2cos0=2"," @y=2sin0=0"," @z="-" 2"," )┤

故(2,0,-2)为所求点的直角坐标.

(2)∵(ρ,θ,z)=(π,π,π),∴{■(x=πcosπ="-" π"," @y=πsinπ=0"," @z=π"," )┤

故(-π,0,π)为所求点的直角坐标.

8把下列用球坐标表示的点用直角坐标表示出来.

(1)(2"," π/6 "," π/3);(2)(2"," π/4 "," 7π/4).

解设点的直角坐标为(x,y,z).

(1)∵(r,φ,θ)=(2"," π/6 "," π/3),

∴{■(x=rsinφcosθ=2sin π/6 cos π/3=1/2 "," @y=rsinφsinθ=2sin π/6 sin π/3=√3/2 "," @z=rcosφ=2cos π/6=√3 "." )┤