Let `vec(a)=(2hat(i)+3hat(j)+2 hat(k))` and `vec(b)=(hat(i)+2hat(j)+hat(k))`.
Find the projection of (i) `vec(a)` on `vec(b)` and (ii) `vec(b)` on `vec(a)`.

Answer

To find the projections of the vectors \(\vec{a}\) and \(\vec{b}\), we will use the formula for the projection of one vector onto another. The projection of vector \(\vec{a}\) onto vector \(\vec{b}\) is given by:

\[ \text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} \]

And the projection of vector \(\vec{b}\) onto vector \(\vec{a}\) is given by:

\[ \text{proj}_{\vec{a}} \vec{b} = \frac{\vec{b} \cdot \vec{a}}{|\vec{a}|^2} \vec{a} \]

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

Given:
\[ \vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k} \]
\[ \vec{b} = \hat{i} + 2\hat{j} + \hat{k} \]

Now, calculate the dot product \(\vec{a} \cdot \vec{b}\):

\[ \vec{a} \cdot \vec{b} = (2)(1) + (3)(2) + (2)(1) = 2 + 6 + 2 = 10 \]

### Step 2: Calculate \(|\vec{b}|^2\)

Now, calculate the magnitude of \(\vec{b}\):

\[ |\vec{b}|^2 = (1)^2 + (2)^2 + (1)^2 = 1 + 4 + 1 = 6 \]

### Step 3: Calculate the projection of \(\vec{a}\) on \(\vec{b}\)

Using the projection formula:

\[ \text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b} = \frac{10}{6} \vec{b} = \frac{5}{3} \vec{b} \]

Substituting \(\vec{b}\):

\[ \text{proj}_{\vec{b}} \vec{a} = \frac{5}{3}(\hat{i} + 2\hat{j} + \hat{k}) = \frac{5}{3}\hat{i} + \frac{10}{3}\hat{j} + \frac{5}{3}\hat{k} \]

### Step 4: Calculate \(|\vec{a}|^2\)

Now, calculate the magnitude of \(\vec{a}\):

\[ |\vec{a}|^2 = (2)^2 + (3)^2 + (2)^2 = 4 + 9 + 4 = 17 \]

### Step 5: Calculate the projection of \(\vec{b}\) on \(\vec{a}\)

Using the projection formula:

\[ \text{proj}_{\vec{a}} \vec{b} = \frac{\vec{b} \cdot \vec{a}}{|\vec{a}|^2} \vec{a} = \frac{10}{17} \vec{a} \]

Substituting \(\vec{a}\):

\[ \text{proj}_{\vec{a}} \vec{b} = \frac{10}{17}(2\hat{i} + 3\hat{j} + 2\hat{k}) = \frac{20}{17}\hat{i} + \frac{30}{17}\hat{j} + \frac{20}{17}\hat{k} \]

### Final Answers

1. The projection of \(\vec{a}\) on \(\vec{b}\) is:
\[ \frac{5}{3}\hat{i} + \frac{10}{3}\hat{j} + \frac{5}{3}\hat{k} \]

2. The projection of \(\vec{b}\) on \(\vec{a}\) is:
\[ \frac{20}{17}\hat{i} + \frac{30}{17}\hat{j} + \frac{20}{17}\hat{k} \]