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

Answer

To find the scalar projection of vector **b** on vector **a**, we can use the formula:

\[ \text{Scalar Projection of } \vec{b} \text{ on } \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|} \]

where \(\vec{a} \cdot \vec{b}\) is the dot product of vectors **a** and **b**, and \(|\vec{a}|\) is the magnitude of vector **a**.

### Step 1: Calculate the dot product \(\vec{a} \cdot \vec{b}\)

Given:
\[ \vec{a} = 2\hat{i} + \hat{j} + 2\hat{k} \]
\[ \vec{b} = \hat{i} + 2\hat{j} + \hat{k} \]

The dot product \(\vec{a} \cdot \vec{b}\) is calculated as follows:
\[ \vec{a} \cdot \vec{b} = (2)(1) + (1)(2) + (2)(1) \]
\[ = 2 + 2 + 2 = 6 \]

### Step 2: Calculate the magnitude of vector \(\vec{a}\)

The magnitude \(|\vec{a}|\) is calculated using the formula:
\[ |\vec{a}| = \sqrt{(2^2) + (1^2) + (2^2)} \]
\[ = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

### Step 3: Calculate the scalar projection

Now we can substitute the values into the scalar projection formula:
\[ \text{Scalar Projection of } \vec{b} \text{ on } \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}|} = \frac{6}{3} = 2 \]

### Final Answer:
The scalar projection of vector **b** on vector **a** is **2**.

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