Alright, let's find the resultant of each statement step-by-step. Given the vectors [tex]\( u = \langle -5, 2 \rangle \)[/tex], [tex]\( v = \langle -1, -3 \rangle \)[/tex], and [tex]\( w = \langle 3, -4 \rangle \)[/tex], we need to find the result of each pair operation and match them with the correct tiles.
1. Pair: [tex]\( -2u + v \)[/tex]
Calculating this:
[tex]\[
-2u = -2 \cdot \langle -5, 2 \rangle = \langle 10, -4 \rangle
\][/tex]
[tex]\[
-2u + v = \langle 10, -4 \rangle + \langle -1, -3 \rangle = \langle 10 - 1, -4 - 3 \rangle = \langle 9, -7 \rangle
\][/tex]
So, the result of [tex]\( -2u + v \)[/tex] is [tex]\( \langle 9, -7 \rangle \)[/tex].
2. Pair: [tex]\( 3v - 3w \)[/tex]
Calculating this:
[tex]\[
3v = 3 \cdot \langle -1, -3 \rangle = \langle -3, -9 \rangle
\][/tex]
[tex]\[
3w = 3 \cdot \langle 3, -4 \rangle = \langle 9, -12 \rangle
\][/tex]
[tex]\[
3v - 3w = \langle -3, -9 \rangle - \langle 9, -12 \rangle = \langle -3 - 9, -9 + 12 \rangle = \langle -12, 3 \rangle
\][/tex]
So, the result of [tex]\( 3v - 3w \)[/tex] is [tex]\( \langle -12, 3 \rangle \)[/tex].
3. Pair: [tex]\( u - v + 2w \)[/tex]
Calculating this:
[tex]\[
-v = -1 \cdot \langle -1, -3 \rangle = \langle 1, 3 \rangle
\][/tex]
[tex]\[
2w = 2 \cdot \langle 3, -4 \rangle = \langle 6, -8 \rangle
\][/tex]
[tex]\[
u - v + 2w = \langle -5, 2 \rangle + \langle 1, 3 \rangle + \langle 6, -8 \rangle = \langle -5 + 1 + 6, 2 + 3 - 8 \rangle = \langle 2, -3 \rangle
\][/tex]
So, the result of [tex]\( u - v + 2w \)[/tex] is [tex]\( \langle 2, -3 \rangle \)[/tex].
4. Pair: [tex]\( 2u - 4w \)[/tex]
Calculating this:
[tex]\[
2u = 2 \cdot \langle -5, 2 \rangle = \langle -10, 4 \rangle
\][/tex]
[tex]\[
4w = 4 \cdot \langle 3, -4 \rangle = \langle 12, -16 \rangle
\][/tex]
[tex]\[
2u - 4w = \langle -10, 4 \rangle - \langle 12, -16 \rangle = \langle -10 - 12, 4 + 16 \rangle = \langle -22, 20 \rangle
\][/tex]
So, the result of [tex]\( 2u - 4w \)[/tex] is [tex]\( \langle -22, 20 \rangle \)[/tex].
Therefore, the correct result pairs are:
- [tex]\( -2u + v \rightarrow \langle 9, -7 \rangle \)[/tex]
- [tex]\( 3v - 3w \rightarrow \langle -12, 3 \rangle \)[/tex]
- [tex]\( u - v + 2w \rightarrow \langle 2, -3 \rangle \)[/tex]
- [tex]\( 2u - 4w \rightarrow \langle -22, 20 \rangle \)[/tex]
