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

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. \( 2u - 5w - \frac{3}{2} v = \langle 5, -15.5 \rangle = 5\mathbf{i} - 15.5\mathbf{j} \)