Three vectors which are coplanar with respect to a certain rectangular co-ordinate system are given by `a = 4 hati - hatj, b = - 3hati + 2hatj and c =- 3hatj`
Find
(a) `a+b+c`
(b) `a+b+-c`
(c ) Find the angle between `a+b+c and a+b-c`

Explanation

To solve the problem step by step, we will follow the instructions given in the question.

### Given Vectors:
- \( \mathbf{a} = 4 \hat{i} - \hat{j} \)
- \( \mathbf{b} = -3 \hat{i} + 2 \hat{j} \)
- \( \mathbf{c} = -3 \hat{j} \)

### (a) Find \( \mathbf{a} + \mathbf{b} + \mathbf{c} \)

1. **Add the vectors component-wise:**
\[ \mathbf{a} + \mathbf{b} + \mathbf{c} = (4 \hat{i} - \hat{j}) + (-3 \hat{i} + 2 \hat{j}) + (0 \hat{i} - 3 \hat{j}) \]

2. **Combine the \( \hat{i} \) components:**
\[ 4 - 3 + 0 = 1 \hat{i} \]

3. **Combine the \( \hat{j} \) components:**
\[ -1 + 2 - 3 = -2 \hat{j} \]

4. **Final result for part (a):**
\[ \mathbf{a} + \mathbf{b} + \mathbf{c} = 1 \hat{i} - 2 \hat{j} \]

### (b) Find \( \mathbf{a} + \mathbf{b} - \mathbf{c} \)

1. **Add the vectors component-wise:**
\[ \mathbf{a} + \mathbf{b} - \mathbf{c} = (4 \hat{i} - \hat{j}) + (-3 \hat{i} + 2 \hat{j}) - (0 \hat{i} - 3 \hat{j}) \]

2. **Combine the \( \hat{i} \) components:**
\[ 4 - 3 + 0 = 1 \hat{i} \]

3. **Combine the \( \hat{j} \) components:**
\[ -1 + 2 + 3 = 4 \hat{j} \]

4. **Final result for part (b):**
\[ \mathbf{a} + \mathbf{b} - \mathbf{c} = 1 \hat{i} + 4 \hat{j} \]

### (c) Find the angle between \( \mathbf{a} + \mathbf{b} + \mathbf{c} \) and \( \mathbf{a} + \mathbf{b} - \mathbf{c} \)

1. **Let \( \mathbf{u} = \mathbf{a} + \mathbf{b} + \mathbf{c} = 1 \hat{i} - 2 \hat{j} \) and \( \mathbf{v} = \mathbf{a} + \mathbf{b} - \mathbf{c} = 1 \hat{i} + 4 \hat{j} \)**

2. **Use the dot product to find the angle:**
\[ \mathbf{u} \cdot \mathbf{v} = (1 \hat{i} - 2 \hat{j}) \cdot (1 \hat{i} + 4 \hat{j}) = 1 \cdot 1 + (-2) \cdot 4 = 1 - 8 = -7 \]

3. **Find the magnitudes of \( \mathbf{u} \) and \( \mathbf{v} \):**
\[ |\mathbf{u}| = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]
\[ |\mathbf{v}| = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \]

4. **Use the formula for the cosine of the angle:**
\[ \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} = \frac{-7}{\sqrt{5} \cdot \sqrt{17}} = \frac{-7}{\sqrt{85}} \]

5. **Find the angle \( \theta \):**
\[ \theta = \cos^{-1}\left(\frac{-7}{\sqrt{85}}\right) \]

### Summary of Results:
- (a) \( \mathbf{a} + \mathbf{b} + \mathbf{c} = 1 \hat{i} - 2 \hat{j} \)
- (b) \( \mathbf{a} + \mathbf{b} - \mathbf{c} = 1 \hat{i} + 4 \hat{j} \)
- (c) \( \theta = \cos^{-1}\left(\frac{-7}{\sqrt{85}}\right) \)