If ` vec a = hati + 2 hatj + 2 hat k and vec b = 3 hati + 6 hatj + 2 hat k ` then the vector in the direction of ` vec a ` and having mgnitude as `|vec b|` is

Options

A. `7( hati + 2 hatj + 2 hat k ) `
B. `(7)/(9)( hati + 2 hatj + 2 hat k ) `
C. `(7)/(3)( hati + 2 hatj + 2 hat k ) `
D. none of these

Answer

To solve the problem, we need to find a vector in the direction of \(\vec{a}\) that has the same magnitude as \(\vec{b}\). Let's break this down step by step.

### Step 1: Define the vectors \(\vec{a}\) and \(\vec{b}\)

Given:
\[ \vec{a} = \hat{i} + 2\hat{j} + 2\hat{k} \]
\[ \vec{b} = 3\hat{i} + 6\hat{j} + 2\hat{k} \]

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

The magnitude of a vector \(\vec{b} = a\hat{i} + b\hat{j} + c\hat{k}\) is given by:
\[ |\vec{b}| = \sqrt{a^2 + b^2 + c^2} \]
For \(\vec{b}\):
\[ |\vec{b}| = \sqrt{3^2 + 6^2 + 2^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

### Step 3: Calculate the magnitude of \(\vec{a}\)

Now, we calculate the magnitude of \(\vec{a}\):
\[ |\vec{a}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

### Step 4: Find the unit vector in the direction of \(\vec{a}\)

The unit vector in the direction of \(\vec{a}\) is given by:
\[ \hat{u}_a = \frac{\vec{a}}{|\vec{a}|} \]
Substituting the values:
\[ \hat{u}_a = \frac{\hat{i} + 2\hat{j} + 2\hat{k}}{3} = \frac{1}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{2}{3}\hat{k} \]

### Step 5: Scale the unit vector to have the magnitude of \(|\vec{b}|\)

To find the vector in the direction of \(\vec{a}\) with magnitude \(|\vec{b}|\), we multiply the unit vector \(\hat{u}_a\) by \(|\vec{b}|\):
\[ \vec{c} = |\vec{b}| \cdot \hat{u}_a = 7 \cdot \left(\frac{1}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{2}{3}\hat{k}\right) \]
Calculating this gives:
\[ \vec{c} = \frac{7}{3}\hat{i} + \frac{14}{3}\hat{j} + \frac{14}{3}\hat{k} \]

### Final Answer

Thus, the vector in the direction of \(\vec{a}\) and having magnitude \(|\vec{b}|\) is:
\[ \vec{c} = \frac{7}{3}\hat{i} + \frac{14}{3}\hat{j} + \frac{14}{3}\hat{k} \]

---