First, let's calculate the sum of the vectors [tex]\( r \)[/tex], [tex]\( s \)[/tex], and [tex]\( t \)[/tex].
[tex]\[
r = (2, 3), \quad s = (5, -3), \quad t = (-8, 6)
\][/tex]
To find [tex]\( r + s + t \)[/tex]:
1. Add the corresponding components of [tex]\( r \)[/tex], [tex]\( s \)[/tex], and [tex]\( t \)[/tex]:
[tex]\[
(r_1 + s_1 + t_1, r_2 + s_2 + t_2) = (2 + 5 + (-8), 3 + (-3) + 6)
\][/tex]
2. Calculate the sums:
[tex]\[
(2 + 5 - 8, 3 - 3 + 6) = (-1, 6)
\][/tex]
Thus, [tex]\( r + s + t = (-1, 6) \)[/tex].
Now, let's check each expression to see which ones sum to the vector [tex]\( (-1, 6) \)[/tex].
1. Expression [tex]\( \langle 0, 7 \rangle + \langle -8, 6 \rangle \)[/tex]
Calculate the sum:
[tex]\[
(0 + (-8), 7 + 6) = (-8, 13)
\][/tex]
This does not match [tex]\( (-1, 6) \)[/tex].
2. Expression [tex]\( \langle 7, 0 \rangle + \langle -8, 6 \rangle \)[/tex]
Calculate the sum:
[tex]\[
(7 + (-8), 0 + 6) = (-1, 6)
\][/tex]
This matches [tex]\( (-1, 6) \)[/tex].
3. Expression [tex]\( \langle 2, 3 \rangle + \langle -3, 3 \rangle \)[/tex]
Calculate the sum:
[tex]\[
(2 + (-3), 3 + 3) = (-1, 6)
\][/tex]
This matches [tex]\( (-1, 6) \)[/tex].
4. Expression [tex]\( \langle 2, 3 \rangle + \langle 3, -3 \rangle \)[/tex]
Calculate the sum:
[tex]\[
(2 + 3, 3 + (-3)) = (5, 0)
\][/tex]
This does not match [tex]\( (-1, 6) \)[/tex].
5. Expression [tex]\( \langle -6, 9 \rangle + \langle 5, -3 \rangle \)[/tex]
Calculate the sum:
[tex]\[
(-6 + 5, 9 + (-3)) = (-1, 6)
\][/tex]
This matches [tex]\( (-1, 6) \)[/tex].
So, the expressions whose sums represent the same vector as [tex]\( (r + s) + t \)[/tex] are:
[tex]\[
\boxed{\langle 7, 0 \rangle + \langle -8, 6 \rangle, \langle 2, 3 \rangle + \langle -3, 3 \rangle, \langle -6, 9 \rangle + \langle 5, -3 \rangle}
\][/tex]
