Sure, let's work through the problem step-by-step.
We are given the vectors [tex]\( w = \langle 3.5, -6 \rangle \)[/tex] and [tex]\( z = \langle -1.5, -4 \rangle \)[/tex]. We need to find the resulting vector for [tex]\( 2w - z \)[/tex].
1. Calculate [tex]\( 2w \)[/tex]:
Let's start by multiplying vector [tex]\( w \)[/tex] by 2:
[tex]\[
2w = 2 \times \langle 3.5, -6 \rangle = \langle 2 \times 3.5, 2 \times (-6) \rangle
\][/tex]
Calculate each component:
[tex]\[
2 \times 3.5 = 7, \quad 2 \times (-6) = -12
\][/tex]
So,
[tex]\[
2w = \langle 7, -12 \rangle
\][/tex]
2. Calculate [tex]\( 2w - z \)[/tex]:
We need to subtract vector [tex]\( z \)[/tex] from [tex]\( 2w \)[/tex]:
[tex]\[
2w - z = \langle 7, -12 \rangle - \langle -1.5, -4 \rangle
\][/tex]
To subtract [tex]\( z \)[/tex], subtract each corresponding component:
[tex]\[
2w - z = \langle 7 - (-1.5), -12 - (-4) \rangle
\][/tex]
Simplify the components:
[tex]\[
7 - (-1.5) = 7 + 1.5 = 8.5
\][/tex]
[tex]\[
-12 - (-4) = -12 + 4 = -8
\][/tex]
3. Resulting vector:
The resulting vector is:
[tex]\[
2w - z = \langle 8.5, -8 \rangle
\][/tex]
So, the correct answer is:
C. [tex]\(\langle 8.5, -8 \rangle\)[/tex]
