Given $$ u=\langle-7,10 angle, v=\langle 4,-3 angle $$, and $$ w=\langle-5,8 angle $$, write each resultant vector in component form and as a linear combination of standard unit vectors $$ i $$ and $$ j $$. $$ \begin{array}{ll} ext { 13. } w+\frac{1}{2} v & ext { 14. } u+v-w \ ext { 15. } 3 v-\frac{1}{2} u+2 w & ext { 16. } 2 u-5 w-\frac{3}{2} v\end{array} $$

Question

Given $$ u=\langle-7,10angle, v=\langle 4,-3angle $$, and $$ w=\langle-5,8angle $$, write each resultant vector in component form and as a linear combination of standard unit vectors $$ i $$ and $$ j $$. $$ \begin{array}{ll}	ext { 13. } w+\frac{1}{2} v & 	ext { 14. } u+v-w \ 	ext { 15. } 3 v-\frac{1}{2} u+2 w & 	ext { 16. } 2 u-5 w-\frac{3}{2} v\end{array} $$

UpStudy AI · Accepted Answer

To solve the problem, we will calculate each resultant vector step by step, expressing them in component form and as a linear combination of the standard unit vectors $$ \mathbf{i} $$ and $$ \mathbf{j} $$.

### Given Vectors
- $$ u = \langle -7, 10 \rangle $$
- $$ v = \langle 4, -3 \rangle $$
- $$ w = \langle -5, 8 \rangle $$

### 13. $$ w + \frac{1}{2} v $$

1. Calculate $$ \frac{1}{2} v $$:
   $$
   \frac{1}{2} v = \frac{1}{2} \langle 4, -3 \rangle = \langle 2, -1.5 \rangle
   $$

2. Add $$ w $$ and $$ \frac{1}{2} v $$:
   $$
   w + \frac{1}{2} v = \langle -5, 8 \rangle + \langle 2, -1.5 \rangle = \langle -5 + 2, 8 - 1.5 \rangle = \langle -3, 6.5 \rangle
   $$

3. Express in terms of $$ \mathbf{i} $$ and $$ \mathbf{j} $$:
   $$
   -3\mathbf{i} + 6.5\mathbf{j}
   $$

### 14. $$ u + v - w $$

1. Add $$ u $$ and $$ v $$:
   $$
   u + v = \langle -7, 10 \rangle + \langle 4, -3 \rangle = \langle -7 + 4, 10 - 3 \rangle = \langle -3, 7 \rangle
   $$

2. Subtract $$ w $$:
   $$
   u + v - w = \langle -3, 7 \rangle - \langle -5, 8 \rangle = \langle -3 + 5, 7 - 8 \rangle = \langle 2, -1 \rangle
   $$

3. Express in terms of $$ \mathbf{i} $$ and $$ \mathbf{j} $$:
   $$
   2\mathbf{i} - 1\mathbf{j}
   $$

### 15. $$ 3v - \frac{1}{2} u + 2w $$

1. Calculate $$ 3v $$:
   $$
   3v = 3 \langle 4, -3 \rangle = \langle 12, -9 \rangle
   $$

2. Calculate $$ \frac{1}{2} u $$:
   $$
   \frac{1}{2} u = \frac{1}{2} \langle -7, 10 \rangle = \langle -3.5, 5 \rangle
   $$

3. Calculate $$ 2w $$:
   $$
   2w = 2 \langle -5, 8 \rangle = \langle -10, 16 \rangle
   $$

4. Combine all:
   $$
   3v - \frac{1}{2} u + 2w = \langle 12, -9 \rangle - \langle -3.5, 5 \rangle + \langle -10, 16 \rangle
   $$
   $$
   = \langle 12 + 3.5 - 10, -9 - 5 + 16 \rangle = \langle 5.5, 2 \rangle
   $$

5. Express in terms of $$ \mathbf{i} $$ and $$ \mathbf{j} $$:
   $$
   5.5\mathbf{i} + 2\mathbf{j}
   $$

### 16. $$ 2u - 5w - \frac{3}{2} v $$

1. Calculate $$ 2u $$:
   $$
   2u = 2 \langle -7, 10 \rangle = \langle -14, 20 \rangle
   $$

2. Calculate $$ 5w $$:
   $$
   5w = 5 \langle -5, 8 \rangle = \langle -25, 40 \rangle
   $$

3. Calculate $$ \frac{3}{2} v $$:
   $$
   \frac{3}{2} v = \frac{3}{2} \langle 4, -3 \rangle = \langle 6, -4.5 \rangle
   $$

4. Combine all:
   $$
   2u - 5w - \frac{3}{2} v = \langle -14, 20 \rangle - \langle -25, 40 \rangle - \langle 6, -4.5 \rangle
   $$
   $$
   = \langle -14 + 25 - 6, 20 - 40 + 4.5 \rangle = \langle 5, -15.5 \rangle
   $$

5. Express in terms of $$ \mathbf{i} $$ and $$ \mathbf{j} $$:
   $$
   5\mathbf{i} - 15.5\mathbf{j}
   $$

### Summary of Results
1. $$ w + \frac{1}{2} v = \langle -3, 6.5 \rangle = -3\mathbf{i} + 6.5\mathbf{j} $$
2. $$ u + v - w = \langle 2, -1 \rangle = 2\mathbf{i} - 1\mathbf{j} $$
3. $$ 3v - \frac{1}{2} u + 2w = \langle 5.5, 2 \rangle = 5.5\mathbf{i} + 2\mathbf{j} $$
4. $$ 2u - 5w - \frac{3}{2} v = \langle 5, -15.5 \rangle = 5\mathbf{i} - 15.5\mathbf{j} $$
1. $$ w + \frac{1}{2} v = \langle -3, 6.5 \rangle = -3\mathbf{i} + 6.5\mathbf{j} $$ 2. $$ u + v - w = \langle 2, -1 \rangle = 2\mathbf{i} - 1\mathbf{j} $$ 3. $$ 3v - \frac{1}{2} u + 2w = \langle 5.5, 2 \rangle = 5.5\mathbf{i} + 2\mathbf{j} $$ 4. $$ 2u - 5w - \frac{3}{2} v = \langle 5, -15.5 \rangle = 5\mathbf{i} - 15.5\mathbf{j} $$

Given , and , write
each resultant vector in component form and as a linear
combination of standard unit vectors and .

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