Find the scalar projection of :
`vec(a)=2hat(i)+3hat(j)+2hat(k)` on `vec(b)=hat(i)+2hat(j)+hat(k)`

Answer

To find the scalar projection of vector \(\vec{a}\) on vector \(\vec{b}\), we will follow these steps:

### Step 1: Identify the vectors
Given:
\[ \vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k} \]
\[ \vec{b} = \hat{i} + 2\hat{j} + \hat{k} \]

### Step 2: Calculate the dot product \(\vec{a} \cdot \vec{b}\)
The dot product of two vectors \(\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) and \(\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\) is given by:
\[ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \]
For our vectors:
\[ \vec{a} \cdot \vec{b} = (2)(1) + (3)(2) + (2)(1) = 2 + 6 + 2 = 10 \]

### Step 3: Calculate the magnitude of vector \(\vec{b}\)
The magnitude of vector \(\vec{b}\) is given by:
\[ |\vec{b}| = \sqrt{b_1^2 + b_2^2 + b_3^2} \]
Calculating for \(\vec{b}\):
\[ |\vec{b}| = \sqrt{(1)^2 + (2)^2 + (1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6} \]

### Step 4: Calculate the scalar projection of \(\vec{a}\) on \(\vec{b}\)
The scalar projection of \(\vec{a}\) on \(\vec{b}\) is given by:
\[ \text{Scalar Projection} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} \]
Substituting the values we found:
\[ \text{Scalar Projection} = \frac{10}{\sqrt{6}} \]

### Final Answer
Thus, the scalar projection of vector \(\vec{a}\) on vector \(\vec{b}\) is:
\[ \frac{10}{\sqrt{6}} \]

---