If `vec a=6hat(i)+7hat(j)+7hat(k)`, find the unit vector along with this vector

Answer

To find the unit vector along the given vector \(\vec{a} = 6\hat{i} + 7\hat{j} + 7\hat{k}\), we will follow these steps:

### Step 1: Identify the vector
The vector is given as:
\[ \vec{a} = 6\hat{i} + 7\hat{j} + 7\hat{k} \]

### Step 2: Calculate the magnitude of the vector
The magnitude of a vector \(\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}\) is calculated using the formula:
\[ |\vec{a}| = \sqrt{x^2 + y^2 + z^2} \]
For our vector:
- \(x = 6\)
- \(y = 7\)
- \(z = 7\)

Now, we calculate the magnitude:
\[ |\vec{a}| = \sqrt{6^2 + 7^2 + 7^2} \]
Calculating each term:
\[ 6^2 = 36, \quad 7^2 = 49, \quad 7^2 = 49 \]
Now, add these values:
\[ |\vec{a}| = \sqrt{36 + 49 + 49} = \sqrt{134} \]

### Step 3: Calculate the unit vector
The unit vector \(\hat{a}\) in the direction of \(\vec{a}\) is given by:
\[ \hat{a} = \frac{\vec{a}}{|\vec{a}|} \]
Substituting the values we have:
\[ \hat{a} = \frac{6\hat{i} + 7\hat{j} + 7\hat{k}}{\sqrt{134}} \]
This can be expressed as:
\[ \hat{a} = \frac{6}{\sqrt{134}}\hat{i} + \frac{7}{\sqrt{134}}\hat{j} + \frac{7}{\sqrt{134}}\hat{k} \]

### Final Result
Thus, the unit vector along \(\vec{a}\) is:
\[ \hat{a} = \frac{6}{\sqrt{134}}\hat{i} + \frac{7}{\sqrt{134}}\hat{j} + \frac{7}{\sqrt{134}}\hat{k} \]

---