To determine the coordinates of point [tex]\(P\)[/tex], let's work through the translations step by step.
1. Understanding Translation Vectors:
- Translation [tex]\(T\)[/tex] is represented by the column vector [tex]\(\binom{5}{4}\)[/tex], which means moving a point 5 units to the right and 4 units up.
- Translation [tex]\(U\)[/tex] is represented by the column vector [tex]\(\binom{-3}{2}\)[/tex], which means moving a point 3 units to the left and 2 units up.
2. Applying Translation [tex]\(U\)[/tex] (Reversely) on [tex]\(R\)[/tex]:
- Point [tex]\(R\)[/tex] has coordinates [tex]\((7, -4)\)[/tex].
- To find the coordinates of point [tex]\(Q\)[/tex] from [tex]\(R\)[/tex], we need to reverse the effects of translation [tex]\(U\)[/tex]. This involves moving [tex]\(R\)[/tex] 3 units to the right and 2 units down.
- Mathematically:
[tex]\[
Q = R - U = (7, -4) - (-3, 2)
\][/tex]
- This simplifies to:
[tex]\[
Q = (7 + 3, -4 - 2) = (10, -6)
\][/tex]
3. Applying Translation [tex]\(T\)[/tex] (Reversely) on [tex]\(Q\)[/tex]:
- Now, point [tex]\(Q\)[/tex] has coordinates [tex]\((10, -6)\)[/tex].
- To find the coordinates of point [tex]\(P\)[/tex] from [tex]\(Q\)[/tex], we need to reverse the effects of translation [tex]\(T\)[/tex]. This involves moving [tex]\(Q\)[/tex] 5 units to the left and 4 units down.
- Mathematically:
[tex]\[
P = Q - T = (10, -6) - (5, 4)
\][/tex]
- This simplifies to:
[tex]\[
P = (10 - 5, -6 - 4) = (5, -10)
\][/tex]
Thus, the coordinates of point [tex]\(P\)[/tex] are [tex]\((5, -10)\)[/tex].
