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A triangular prism has vertices at A(2,-1,-1) B(2,1,4) C(2,2,-1) D(-1,-1,-1) E(-1,1,4), and F(-1,2,-1)?
This question has two parts:
1 - Which image point has the coordinates (-3,2,1) after a translation using the vector <-5,1,3>
I'm thinking it's point B.
2 - What point represents a reflection of B over the yz plane?
Of the multiple choice, the four answers did not provide the one I came up with. (I had B'(-2,2,4).) They gave me
A - (-2,-1,4)
B - (-2,1,4)
C - (-2,2,-4)
D - (-2,1,-4)
Which one should it be and why?
1 - To determine which image point has the coordinates (-3,2,1) after a translation using the vector <-5,1,3>, you can apply the vector translation formula. The formula is:
P' = P + v
where P' is the new position of the point after translation, P is the original position of the point, and v is the translation vector.
For each of the given points A, B, C, D, E, and F, you can add the translation vector <-5,1,3> to the coordinates of the respective point to obtain the new position.
For point A(2,-1,-1):
A' = A + <-5,1,3> = (2-5, -1+1, -1+3) = (-3, 0, 2)
For point B(2,1,4):
B' = B + <-5,1,3> = (2-5, 1+1, 4+3) = (-3, 2, 7)
For point C(2,2,-1):
C' = C + <-5,1,3> = (2-5, 2+1, -1+3) = (-3, 3, 2)
For point D(-1,-1,-1):
D' = D + <-5,1,3> = (-1-5, -1+1, -1+3) = (-6, 0, 2)
For point E(-1,1,4):
E' = E + <-5,1,3> = (-1-5, 1+1, 4+3) = (-6, 2, 7)
For point F(-1,2,-1):
F' = F + <-5,1,3> = (-1-5, 2+1, -1+3) = (-6, 3, 2)
As you can see, the point with the coordinates (-3, 2, 1) after the translation using the vector <-5,1,3> is B'.
2 - To find the reflection of point B over the yz plane, you need to negate the x-coordinate of point B while keeping the y and z coordinates the same.
Original point B(2, 1, 4)
Reflection point B'(-2, 1, 4)
From the options given:
A - (-2,-1,4): This is not the reflection of point B because the y-coordinate has changed.
B - (-2,1,4): This is the correct reflection of point B.
C - (-2,2,-4): This is not the reflection of point B because the z-coordinate has changed.
D - (-2,1,-4): This is not the reflection of point B because both the y and z coordinates have changed.
Therefore, the correct answer is B (-2,1,4) for the reflection of point B over the yz plane.