Find the unit vector of the vector `vec(r ) = 4hat(i) - 2hat(j) + 3 hat(k)`

Explanation

To find the unit vector of the vector \(\vec{r} = 4\hat{i} - 2\hat{j} + 3\hat{k}\), we will follow these steps:

### Step 1: Identify the vector components
The vector \(\vec{r}\) can be expressed in terms of its components:
- \(x\) component: \(4\)
- \(y\) component: \(-2\)
- \(z\) component: \(3\)

### Step 2: Calculate the magnitude of the vector
The magnitude of the vector \(\vec{r}\) is calculated using the formula:
\[ |\vec{r}| = \sqrt{x^2 + y^2 + z^2} \]
Substituting the values of the components:
\[ |\vec{r}| = \sqrt{4^2 + (-2)^2 + 3^2} \]
Calculating each term:
\[ |\vec{r}| = \sqrt{16 + 4 + 9} \]
\[ |\vec{r}| = \sqrt{29} \]

### Step 3: Calculate the unit vector
The unit vector \(\hat{r}\) in the direction of \(\vec{r}\) is given by:
\[ \hat{r} = \frac{\vec{r}}{|\vec{r}|} \]
Substituting \(\vec{r}\) and its magnitude:
\[ \hat{r} = \frac{4\hat{i} - 2\hat{j} + 3\hat{k}}{\sqrt{29}} \]
This can be expressed as:
\[ \hat{r} = \frac{4}{\sqrt{29}}\hat{i} - \frac{2}{\sqrt{29}}\hat{j} + \frac{3}{\sqrt{29}}\hat{k} \]

### Final Answer
Thus, the unit vector of \(\vec{r}\) is:
\[ \hat{r} = \frac{4}{\sqrt{29}}\hat{i} - \frac{2}{\sqrt{29}}\hat{j} + \frac{3}{\sqrt{29}}\hat{k} \]

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