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The eight vertices of a rectangular prism are as follows, Find the coordinates of the vertices after the prism is rotated counterclockwise about the z- axis.


Question 

The eight vertices of a rectangular prism are as follows:

V1=(0,0,0),V2=(1,0,0),V3=(1,2,0),V4=(0,2,0)

V5=(0,0,3),V6=(1,0,3),V7=(1,2,3),V8=(0,2,3)

Find the coordinates of the vertices after the prism is rotated counterclockwise about the z- axis through θ =60∘.





Solution

Here  θ =60

and

The prism is rotated counterclockwise about the z -axis 

then

now we first find V- 1=(x-1,y-1,z-1)

Here   V1=(0,0,0)

then

V- 1=(0,0,0)

Now  V- 2= =(x-2,y-2,z-2)

here    V2=(1,0,0)

                x2=1,y2=0,andz2=0

x- 2=x2cos60-y2sin60=(1)(12)-(0)(√32)=12

y- 2=x2sin60+y2cos60=(1)(√32)+(0)(12)=√32

z- 2=z2=0

So,

V- 2=(12,√32,0)

Now

V- 3=(x-3,y-3,z-3)

here

V3=(1,2,0)

x3=1,y3=2,z3=0

x- 3=x3cos60-y3sin60=(1)(12)-(2)(√32)=12-√3

y- 3=x3sin60+y3cos60=(1)(√32)+(2)(12)=√32-1

z- 3=z3=0

So,

V- 3=(12-√3,√32-1,0)

Now

V- 4=(x-4,y-4,z-4)

V4=(0,2,0)

x4=0,y4=2,z4=0

x- 4=x4cos60-y4sin60=(0)(12)-(2)(√32)= -√3

y- 4=x4sin60+y4cos60=(0)(√32)+(2)(12)= -1

z- 4=z4=0

So,

V- 4=(-√3,-1,0)

Now V- 5=(x-5,y-5,z-5)

here

V5=(0,0,3)

x5=0,y5=0,z5=3

x- 5=x5cos60-y5sin60=(0)(12)-(0)(√32)=0

y- 5=x5sin60+y5cos60=(0)(√32)+(0)(12)=0

z- 5=z5=3

So,

V- 5=(0,0,3)

Now

V- 6=(x-6,y-6,z-6)

here  V6=(1,0,3)

x6=1,y6=0,z6=3

x- 6=x6cos60-y6sin60=(1)(12)-(0)(√32)=12

y- 6=x6sin60+y6cos60=(1)(√32)+(0)(12)=√32

z- 6=z6=3

So,

 

V- 6=(12,√32,3)

 

Now V- 7=(x-7,y-7,z-7)

here V7=(1,2,3)

x7=1,y7=2,z7=3

x- 7=x7cos60-x7sin60=(1)(12)-(2)(√32)=12-√3

y- 7=x7sin60+x7cos60=(1)(√32)+(2)(12)=√32-1

z- 7=z7=3

So,

V- 7=(12-√3,√32-,3)

 

Now V- 8=(x-8,y-8,z-8)

here V8=(0,2,3)

x8=0, y8=2,andz8=3

x- 8=x8cos60-y8sin60=(0)(12)-(2)(√32)= -√3

y- 8=x8sin60+y8cos60=(0)(√32)+(2)(12)= -1

z- 8=z8=3

So,

V- 8=(-√3,-1,3)

\]

Hence,

The coordinates of the vertices after the prism are rotated counterclockwise about the z-axis through θ =60∘ are

V- 1=(0,0,0),V- 2=(12,√32,0),

V- 3=(12-√3,√32-1,0),V- 4=(-√3,-1,0), 

V- 5=(0,0,3),V- 6=(12,√32,3),

V- 7=(12-√3,√32-,3),V- 8=(-√3,-1,3)




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