The projection of the vector `vec( a) = 2 hat(i) - hat(j) +hat(k)` along the vector `vec(b) = hat(i) + 2 hat(j)+ 2hat(k)` is

Options

A. 2
B. `sqrt(6)`
C. `(2)/(3)`
D. `(1)/(3)`

Answer

To find the projection of the vector \(\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}\) along the vector \(\vec{b} = \hat{i} + 2 \hat{j} + 2 \hat{k}\), we can use the formula for the projection of vector \(\vec{a}\) onto vector \(\vec{b}\):

\[ \text{Projection of } \vec{a} \text{ on } \vec{b} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} \]

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

\[ \vec{a} \cdot \vec{b} = (2 \hat{i} - \hat{j} + \hat{k}) \cdot (\hat{i} + 2 \hat{j} + 2 \hat{k}) \]

Calculating the dot product:

\[ = 2 \cdot 1 + (-1) \cdot 2 + 1 \cdot 2 \]
\[ = 2 - 2 + 2 = 2 \]

### Step 2: Calculate the magnitude squared of \(\vec{b}\)

\[ |\vec{b}|^2 = (\hat{i} + 2 \hat{j} + 2 \hat{k}) \cdot (\hat{i} + 2 \hat{j} + 2 \hat{k}) \]

Calculating the magnitude squared:

\[ = 1^2 + 2^2 + 2^2 = 1 + 4 + 4 = 9 \]

### Step 3: Substitute the values into the projection formula

Now we can substitute the values we found into the projection formula:

\[ \text{Projection of } \vec{a} \text{ on } \vec{b} = \frac{2}{9} \vec{b} \]

### Step 4: Write the final projection vector

Substituting \(\vec{b}\):

\[ = \frac{2}{9} (\hat{i} + 2 \hat{j} + 2 \hat{k}) = \frac{2}{9} \hat{i} + \frac{4}{9} \hat{j} + \frac{4}{9} \hat{k} \]

Thus, the projection of the vector \(\vec{a}\) along the vector \(\vec{b}\) is:

\[ \frac{2}{9} \hat{i} + \frac{4}{9} \hat{j} + \frac{4}{9} \hat{k} \]