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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 two parts, we will use vector operations and transformations. Let's start with the first part:
1) Translation using the vector <-5,1,3>:
To translate a point P(x,y,z) by a vector <a,b,c>, we can use the formula P' = P + <a,b,c>.
Let's apply this to the given point (-3,2,1) and the translation vector <-5,1,3>.
P' = (-3,2,1) + <-5,1,3>
To add two vectors, we add their corresponding components:
P' = (-3-5, 2+1, 1+3)
= (-8, 3, 4)
Therefore, the image point after the translation is (-8,3,4).
Now let's move on to the second part:
2) Reflection over the yz plane:
To reflect a point P(x,y,z) over the yz plane, we need to change the sign of the x-coordinate. So if we have a point B(x,y,z), its reflection over the yz plane will be B'(-x,y,z).
Given that the original point B has coordinates (2,1,4), we can find its reflection B' as B'(-2,1,4).
Comparing this with the given options:
A - (-2,-1,4) --> Incorrect since only the y-coordinate is negated.
B - (-2,1,4) --> Correct answer since only the x-coordinate is negated.
C - (-2,2,-4) --> Incorrect since both the x and z coordinates are negated.
D - (-2,1,-4) --> Incorrect since only the x-coordinate is negated.
Therefore, the correct answer for the reflection of B over the yz plane is B'(-2,1,4).
I hope this clarifies your doubts and helps you understand how to solve similar problems in the future!