$$ \left. \begin{array} { l } { \vec { u } = ( - 7,8 ) } \ { \vec { w } = ( - 5 , - 1 ) } \ { \frac { 1 } { 2 } \vec { u } - 2 \vec { w } = ( \square , \square ) } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { \vec { u } = ( - 7,8 ) } \ { \vec { w } = ( - 5 , - 1 ) } \ { \frac { 1 } { 2 } \vec { u } - 2 \vec { w } = ( \square , \square ) } \end{array} ight. $$

UpStudy AI · Accepted Answer

Given 
$$
\vec{u} = (-7,8) \quad \text{and} \quad \vec{w} = (-5,-1),
$$
we want to calculate 
$$
\frac{1}{2}\vec{u} - 2\vec{w}.
$$

1. First, compute $$\frac{1}{2}\vec{u}$$:
   $$
   \frac{1}{2}\vec{u} = \left(\frac{1}{2} \cdot (-7), \frac{1}{2} \cdot 8\right) = \left(-\frac{7}{2}, 4\right).
   $$

2. Next, compute $$-2\vec{w}$$:
   $$
   -2\vec{w} = -2 \cdot (-5,-1) = \left(-2 \cdot (-5), -2 \cdot (-1)\right) = (10,2).
   $$

3. Now, add the results:
   $$
   \frac{1}{2}\vec{u} - 2\vec{w} = \left(-\frac{7}{2}, 4\right) + (10,2) = \left(-\frac{7}{2} + 10,\, 4 + 2\right).
   $$
   
   Convert $$10$$ to a fraction with denominator 2:
   $$
   10 = \frac{20}{2}.
   $$
   
   Therefore, the $$x$$-coordinate is:
   $$
   -\frac{7}{2} + \frac{20}{2} = \frac{13}{2}.
   $$

4. The $$y$$-coordinate is:
   $$
   4 + 2 = 6.
   $$

Thus, the final answer is:
$$
\frac{1}{2}\vec{u} - 2\vec{w} = \left(\frac{13}{2}, 6\right).
$$
$$ \frac{1}{2}\vec{u} - 2\vec{w} = \left(\frac{13}{2}, 6\right) $$

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