To find the composition of translations [tex]\( T_{\langle 7, 8 \rangle} \circ T_{\langle -3, -4 \rangle} \)[/tex] as a single translation, you need to combine the translations by adding their corresponding components together.
1. Consider the first translation vector [tex]\( T_{\langle 7, 8 \rangle} \)[/tex]:
[tex]\[
\text{Translation 1} = (7, 8)
\][/tex]
2. Consider the second translation vector [tex]\( T_{\langle -3, -4 \rangle} \)[/tex]:
[tex]\[
\text{Translation 2} = (-3, -4)
\][/tex]
3. To find the resulting translation, add the corresponding components of these two vectors:
[tex]\[
\begin{aligned}
x\text{-component:}\ & 7 + (-3) = 4, \\
y\text{-component:}\ & 8 + (-4) = 4.
\end{aligned}
\][/tex]
Therefore, the resulting translation vector from the composition [tex]\( T_{\langle 7, 8 \rangle} \circ T_{\langle -3, -4 \rangle} \)[/tex] is:
[tex]\[
T_{\langle 4, 4 \rangle}
\][/tex]
So, the composition of the translations [tex]\( T_{\langle 7, 8 \rangle} \circ T_{\langle -3, -4 \rangle} \)[/tex] as one translation is:
\[
T_{\langle 4, 4 \rangle}
\
