11. Find the terminal points for each vector. (a) \( \boldsymbol{w}=4 \boldsymbol{i}+2 \boldsymbol{j}-2 \boldsymbol{k} \), given the initial point \( (-1,2,-3) \) (b) \( \boldsymbol{v}=2 \boldsymbol{i}-3 \boldsymbol{j}+\boldsymbol{k} \), given the initial point \( (-2,1,4) \)

To find the terminal points for each vector given an initial point, we can use the formula: \[ \text{Terminal Point} = \text{Initial Point} + \text{Vector} \] Where the vector is represented in component form. Let's solve each part step by step. ### Part (a) Given: - Vector \( \boldsymbol{w} = 4 \boldsymbol{i} + 2 \boldsymbol{j} - 2 \boldsymbol{k} \) - Initial point \( (-1, 2, -3) \) 1. **Convert the vector to component form**: \[ \boldsymbol{w} = (4, 2, -2) \] 2. **Add the vector to the initial point**: \[ \text{Terminal Point} = (-1, 2, -3) + (4, 2, -2) \] 3. **Calculate each component**: - \( x \)-component: \( -1 + 4 = 3 \) - \( y \)-component: \( 2 + 2 = 4 \) - \( z \)-component: \( -3 - 2 = -5 \) 4. **Combine the components**: \[ \text{Terminal Point} = (3, 4, -5) \] ### Part (b) Given: - Vector \( \boldsymbol{v} = 2 \boldsymbol{i} - 3 \boldsymbol{j} + \boldsymbol{k} \) - Initial point \( (-2, 1, 4) \) 1. **Convert the vector to component form**: \[ \boldsymbol{v} = (2, -3, 1) \] 2. **Add the vector to the initial point**: \[ \text{Terminal Point} = (-2, 1, 4) + (2, -3, 1) \] 3. **Calculate each component**: - \( x \)-component: \( -2 + 2 = 0 \) - \( y \)-component: \( 1 - 3 = -2 \) - \( z \)-component: \( 4 + 1 = 5 \) 4. **Combine the components**: \[ \text{Terminal Point} = (0, -2, 5) \] ### Final Answers - (a) The terminal point for vector \( \boldsymbol{w} \) is \( (3, 4, -5) \). - (b) The terminal point for vector \( \boldsymbol{v} \) is \( (0, -2, 5) \).