If `vec(a) = 2 hat(i) + hat(j) + 2hat(k)` and `vec(b) = 5hat(i)- 3 hat(j) + hat(k)` , then the projection of `vec(b)` on ` vec(a)` is

Options

A. 3
B. 4
C. 5
D. 1

Answer

To find the projection of vector **b** on vector **a**, we will use the formula for projection:

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

### Step 1: Define the vectors
Given:
\[ \vec{a} = 2\hat{i} + \hat{j} + 2\hat{k} \]
\[ \vec{b} = 5\hat{i} - 3\hat{j} + \hat{k} \]

### Step 2: Calculate the dot product \(\vec{b} \cdot \vec{a}\)
The dot product is calculated as follows:
\[ \vec{b} \cdot \vec{a} = (5\hat{i} - 3\hat{j} + \hat{k}) \cdot (2\hat{i} + \hat{j} + 2\hat{k}) \]
Using the properties of dot product:
\[ = 5 \cdot 2 + (-3) \cdot 1 + 1 \cdot 2 \]
Calculating each term:
\[ = 10 - 3 + 2 = 9 \]

### Step 3: Calculate the magnitude of vector \(\vec{a}\)
The magnitude of vector \(\vec{a}\) is given by:
\[ |\vec{a}| = \sqrt{(2)^2 + (1)^2 + (2)^2} \]
Calculating:
\[ = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]

### Step 4: Calculate the projection
Now, substituting the values into the projection formula:
\[ \text{Projection of } \vec{b} \text{ on } \vec{a} = \frac{9}{3} = 3 \]

### Final Answer
The projection of vector **b** on vector **a** is:
\[ \boxed{3} \]