Write a unit vector in the direction of the sum of the vectors : `vec(a)=2hat(i)+2hat(j)-5hat(k)` and `vec(b)=2hat(i)+hat(j)+3hat(k)`.

Answer

To find a unit vector in the direction of the sum of the vectors \(\vec{a} = 2\hat{i} + 2\hat{j} - 5\hat{k}\) and \(\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}\), we will follow these steps:

### Step 1: Find the sum of the vectors \(\vec{a}\) and \(\vec{b}\).

\[ \vec{r} = \vec{a} + \vec{b} \]

Substituting the values of \(\vec{a}\) and \(\vec{b}\):

\[ \vec{r} = (2\hat{i} + 2\hat{j} - 5\hat{k}) + (2\hat{i} + \hat{j} + 3\hat{k}) \]

### Step 2: Combine like terms.

\[ \vec{r} = (2 + 2)\hat{i} + (2 + 1)\hat{j} + (-5 + 3)\hat{k} \]

Calculating each component:

\[ \vec{r} = 4\hat{i} + 3\hat{j} - 2\hat{k} \]

### Step 3: Calculate the magnitude of the vector \(\vec{r}\).

The magnitude \(|\vec{r}|\) is given by:

\[ |\vec{r}| = \sqrt{(4)^2 + (3)^2 + (-2)^2} \]

Calculating the squares:

\[ |\vec{r}| = \sqrt{16 + 9 + 4} = \sqrt{29} \]

### Step 4: Find the unit vector in the direction of \(\vec{r}\).

The unit vector \(\hat{r}\) in the direction of \(\vec{r}\) is given by:

\[ \hat{r} = \frac{\vec{r}}{|\vec{r}|} \]

Substituting the values:

\[ \hat{r} = \frac{4\hat{i} + 3\hat{j} - 2\hat{k}}{\sqrt{29}} \]

### Step 5: Write the final expression for the unit vector.

\[ \hat{r} = \frac{4}{\sqrt{29}} \hat{i} + \frac{3}{\sqrt{29}} \hat{j} - \frac{2}{\sqrt{29}} \hat{k} \]

Thus, the unit vector in the direction of the sum of the vectors \(\vec{a}\) and \(\vec{b}\) is:

\[ \hat{r} = \frac{4}{\sqrt{29}} \hat{i} + \frac{3}{\sqrt{29}} \hat{j} - \frac{2}{\sqrt{29}} \hat{k} \]

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