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.

### Step 1: Find \( a + b + c \)

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

Now, we will add these vectors together:

\[ \mathbf{a} + \mathbf{b} + \mathbf{c} = (4 \hat{i} - \hat{j}) + (-3 \hat{i} + 2 \hat{j}) + (-3 \hat{j}) \]

Combine the \( \hat{i} \) components and \( \hat{j} \) components:

\[ = (4 - 3) \hat{i} + (-1 + 2 - 3) \hat{j} \]
\[ = 1 \hat{i} - 2 \hat{j} \]

So,

\[ \mathbf{a} + \mathbf{b} + \mathbf{c} = \hat{i} - 2 \hat{j} \]

### Step 2: Find \( a + b - c \)

Now, we will calculate \( \mathbf{a} + \mathbf{b} - \mathbf{c} \):

\[ \mathbf{a} + \mathbf{b} - \mathbf{c} = (4 \hat{i} - \hat{j}) + (-3 \hat{i} + 2 \hat{j}) - (-3 \hat{j}) \]

This simplifies to:

\[ = (4 - 3) \hat{i} + (-1 + 2 + 3) \hat{j} \]
\[ = 1 \hat{i} + 4 \hat{j} \]

So,

\[ \mathbf{a} + \mathbf{b} - \mathbf{c} = \hat{i} + 4 \hat{j} \]

### Step 3: Find the angle between \( a + b + c \) and \( a + b - c \)

Let \( \mathbf{R_1} = \mathbf{a} + \mathbf{b} + \mathbf{c} = \hat{i} - 2 \hat{j} \) and \( \mathbf{R_2} = \mathbf{a} + \mathbf{b} - \mathbf{c} = \hat{i} + 4 \hat{j} \).

To find the angle \( \theta \) between the two vectors, we use the dot product formula:

\[ \mathbf{R_1} \cdot \mathbf{R_2} = |\mathbf{R_1}| |\mathbf{R_2}| \cos \theta \]

Calculating the dot product:

\[ \mathbf{R_1} \cdot \mathbf{R_2} = (1)(1) + (-2)(4) = 1 - 8 = -7 \]

Now, we need to find the magnitudes of \( \mathbf{R_1} \) and \( \mathbf{R_2} \):

\[ |\mathbf{R_1}| = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

\[ |\mathbf{R_2}| = \sqrt{(1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17} \]

Now substituting back into the dot product equation:

\[ -7 = \sqrt{5} \cdot \sqrt{17} \cdot \cos \theta \]

Calculating \( \sqrt{5} \cdot \sqrt{17} = \sqrt{85} \):

\[ -7 = \sqrt{85} \cdot \cos \theta \]

Thus,

\[ \cos \theta = \frac{-7}{\sqrt{85}} \]

Finally, to find \( \theta \):

\[ \theta = \cos^{-1} \left( \frac{-7}{\sqrt{85}} \right) \]

### Summary of Solutions:
(a) \( \mathbf{a} + \mathbf{b} + \mathbf{c} = \hat{i} - 2 \hat{j} \)

(b) \( \mathbf{a} + \mathbf{b} - \mathbf{c} = \hat{i} + 4 \hat{j} \)

(c) \( \theta = \cos^{-1} \left( \frac{-7}{\sqrt{85}} \right) \)