To solve this problem, we need to find a vector [tex]\( w \)[/tex] that can be added to the vector [tex]\( u + 3v \)[/tex] to get the result to be the unit vector [tex]\( \langle 1, 0 \rangle \)[/tex].
Step 1: Calculate the vector [tex]\( u + 3v \)[/tex].
Given:
[tex]\[ u = \langle -7, 6 \rangle \][/tex]
[tex]\[ v = \langle -4, 17 \rangle \][/tex]
First, we compute [tex]\( 3v \)[/tex]:
[tex]\[ 3v = 3 \times \langle -4, 17 \rangle = \langle 3 \times -4, 3 \times 17 \rangle = \langle -12, 51 \rangle \][/tex]
Now, add [tex]\( u \)[/tex] and [tex]\( 3v \)[/tex]:
[tex]\[ u + 3v = \langle -7, 6 \rangle + \langle -12, 51 \rangle = \langle -7 + -12, 6 + 51 \rangle = \langle -19, 57 \rangle \][/tex]
Step 2: Determine the vector [tex]\( w \)[/tex] such that [tex]\( (u + 3v) + w = \langle 1, 0 \rangle \)[/tex].
We need:
[tex]\[ \langle -19, 57 \rangle + w = \langle 1, 0 \rangle \][/tex]
Let [tex]\( w = \langle x, y \rangle \)[/tex]. Then,
[tex]\[ \langle -19 + x, 57 + y \rangle = \langle 1, 0 \rangle \][/tex]
This gives us two equations:
[tex]\[ -19 + x = 1 \][/tex]
[tex]\[ 57 + y = 0 \][/tex]
Solving these equations:
[tex]\[ x = 1 + 19 = 20 \][/tex]
[tex]\[ y = -57 \][/tex]
Thus, [tex]\( w = \langle 20, -57 \rangle \)[/tex].
Therefore, the correct answer is:
C. [tex]\( w = \langle 20, -57 \rangle \)[/tex]
