Find the vector equation of a sphere with centre having the position vector `hat(i)+hat(j)+hat(k) and sqrt(3).`

Answer

To find the vector equation of a sphere with a given center and radius, we can follow these steps:

### Step 1: Identify the center and radius
The center of the sphere is given by the position vector \( \mathbf{a} = \hat{i} + \hat{j} + \hat{k} \) and the radius \( r = \sqrt{3} \).

### Step 2: Write the formula for the vector equation of a sphere
The vector equation of a sphere can be expressed as:
\[ |\mathbf{r} - \mathbf{a}| = r \]
where \( \mathbf{r} \) is the position vector of any point on the sphere, \( \mathbf{a} \) is the position vector of the center, and \( r \) is the radius.

### Step 3: Substitute the values into the equation
Substituting the values we have:
- \( \mathbf{a} = \hat{i} + \hat{j} + \hat{k} \)
- \( r = \sqrt{3} \)

The equation becomes:
\[ |\mathbf{r} - (\hat{i} + \hat{j} + \hat{k})| = \sqrt{3} \]

### Step 4: Finalize the vector equation of the sphere
Thus, the vector equation of the sphere is:
\[ |\mathbf{r} - \hat{i} - \hat{j} - \hat{k}| = \sqrt{3} \]

This is the required vector equation of the sphere.

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