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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 reflaction 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?
To solve these questions, we need to understand the concepts of translation and reflection in 3D space.
1) For the first question, we are given a triangular prism with vertices 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). We want to find which image point has the coordinates (-3,2,1) after a translation using the vector <-5,1,3>.
To perform a translation, we add the components of the translation vector to the corresponding coordinates of the original points. So, to find the new coordinates after translation, we add the corresponding components of the translation vector <-5,1,3> to the coordinates of each point.
For point B(2,1,4):
New x-coordinate = 2 + (-5) = -3
New y-coordinate = 1 + 1 = 2
New z-coordinate = 4 + 3 = 7
The new coordinates of point B after translation are (-3, 2, 7), not (-3, 2, 1). Therefore, point B is not the correct answer. You'll need to recheck your calculations or identify the correct point.
2) For the second question, we want to find the reflection of point B(2,1,4) over the yz-plane. A reflection over a plane involves changing the sign of one of the coordinates while leaving the others unchanged.
The yz-plane is defined by the equation x = 0, which means the x-coordinate of any point on the plane is 0. So, to find the reflection of point B, we need to change the sign of its x-coordinate while leaving the y and z coordinates unchanged.
So the reflection of B(2,1,4) over the yz-plane would have the coordinates:
x-coordinate = -2
y-coordinate = 1 (unchanged)
z-coordinate = 4 (unchanged)
Among the options provided, the correct point representing the reflection of B over the yz-plane is B'(-2,1,4). None of the options A, C, or D match the correct answer.