To find the composition of the translations [tex]\(T_{\langle 0,3\rangle} \circ T_{\langle 4,6\rangle}\)[/tex], we need to combine the effects of these translations into a single translation vector.
Let's denote the translation vectors as follows:
- [tex]\( T_{\langle 0, 3 \rangle} = \langle 0, 3 \rangle \)[/tex]
- [tex]\( T_{\langle 4, 6 \rangle} = \langle 4, 6 \rangle \)[/tex]
To compose these translations, we add their corresponding components. This means we add the x-components together and the y-components together:
1. For the x-component:
[tex]\[
0 + 4 = 4
\][/tex]
2. For the y-component:
[tex]\[
3 + 6 = 9
\][/tex]
So, the resulting translation vector is:
[tex]\[
T_{\langle 4, 9 \rangle}
\][/tex]
Therefore, the composition of the translations [tex]\(T_{\langle 0,3\rangle} \circ T_{\langle 4,6\rangle}\)[/tex] is [tex]\(T_{\langle 4, 9 \rangle}\)[/tex].
