a. Calculate `vec(r )=vec(a)-vec(b)+vec(c )`, where `vec(a)=5hat(i)+4hat(j)-6hat(k)`,
`vec(b)= -2hat(i)+2hat(j)+3hat(k)`, and `vec(c )= 4hat(i)+3hat(j)+2hat(k)`.
b. Calculate the angle between `vec(r )` and the `z`-axis.

Explanation

To solve the problem step by step, we will first calculate the vector \(\vec{r}\) using the given vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). Then, we will find the angle between \(\vec{r}\) and the \(z\)-axis.

### Part a: Calculate \(\vec{r} = \vec{a} - \vec{b} + \vec{c}\)

1. **Write down the given vectors:**
- \(\vec{a} = 5\hat{i} + 4\hat{j} - 6\hat{k}\)
- \(\vec{b} = -2\hat{i} + 2\hat{j} + 3\hat{k}\)
- \(\vec{c} = 4\hat{i} + 3\hat{j} + 2\hat{k}\)

2. **Calculate \(\vec{a} - \vec{b}\):**
\[ \vec{a} - \vec{b} = (5\hat{i} + 4\hat{j} - 6\hat{k}) - (-2\hat{i} + 2\hat{j} + 3\hat{k}) \]
\[ = (5 + 2)\hat{i} + (4 - 2)\hat{j} + (-6 - 3)\hat{k} \]
\[ = 7\hat{i} + 2\hat{j} - 9\hat{k} \]

3. **Now add \(\vec{c}\):**
\[ \vec{r} = (7\hat{i} + 2\hat{j} - 9\hat{k}) + (4\hat{i} + 3\hat{j} + 2\hat{k}) \]
\[ = (7 + 4)\hat{i} + (2 + 3)\hat{j} + (-9 + 2)\hat{k} \]
\[ = 11\hat{i} + 5\hat{j} - 7\hat{k} \]

Thus, \(\vec{r} = 11\hat{i} + 5\hat{j} - 7\hat{k}\).

### Part b: Calculate the angle between \(\vec{r}\) and the \(z\)-axis

1. **Identify the \(z\)-component of \(\vec{r}\):**
- The \(z\)-component of \(\vec{r}\) is \(-7\).

2. **Calculate the magnitude of \(\vec{r}\):**
\[ |\vec{r}| = \sqrt{(11)^2 + (5)^2 + (-7)^2} \]
\[ = \sqrt{121 + 25 + 49} \]
\[ = \sqrt{195} \]

3. **Use the cosine formula to find the angle \(\gamma\) between \(\vec{r}\) and the \(z\)-axis:**
\[ \cos \gamma = \frac{\text{z-component of } \vec{r}}{|\vec{r}|} \]
\[ \cos \gamma = \frac{-7}{\sqrt{195}} \]

4. **Calculate the angle \(\gamma\):**
\[ \gamma = \cos^{-1}\left(\frac{-7}{\sqrt{195}}\right) \]

### Final Answers:
- \(\vec{r} = 11\hat{i} + 5\hat{j} - 7\hat{k}\)
- \(\gamma = \cos^{-1}\left(\frac{-7}{\sqrt{195}}\right)\)