If `a=hati+2hatj+2hatk and b=3hati+6hatj+2hatk`, then find a vector in the direction of a and having magnitude as |b|.

Answer

To solve the problem, we need to find a vector in the direction of vector **a** with a magnitude equal to the magnitude of vector **b**. Let's break down the solution step by step.

### Step 1: Define the vectors
Given:
\[ \mathbf{a} = \hat{i} + 2\hat{j} + 2\hat{k} \]
\[ \mathbf{b} = 3\hat{i} + 6\hat{j} + 2\hat{k} \]

### Step 2: Calculate the magnitudes of vectors **a** and **b**
To find the magnitude of vector **a**:
\[ |\mathbf{a}| = \sqrt{(1)^2 + (2)^2 + (2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

To find the magnitude of vector **b**:
\[ |\mathbf{b}| = \sqrt{(3)^2 + (6)^2 + (2)^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

### Step 3: Find the unit vector in the direction of **a**
The unit vector in the direction of **a** is given by:
\[ \hat{a} = \frac{\mathbf{a}}{|\mathbf{a}|} = \frac{\hat{i} + 2\hat{j} + 2\hat{k}}{3} = \frac{1}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{2}{3}\hat{k} \]

### Step 4: Scale the unit vector to have a magnitude equal to |b|
To find the vector in the direction of **a** with a magnitude equal to |b|, we multiply the unit vector by the magnitude of **b**:
\[ \mathbf{v} = |\mathbf{b}| \cdot \hat{a} = 7 \left(\frac{1}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{2}{3}\hat{k}\right) \]
\[ \mathbf{v} = \frac{7}{3}\hat{i} + \frac{14}{3}\hat{j} + \frac{14}{3}\hat{k} \]

### Final Answer
Thus, the required vector in the direction of **a** with a magnitude equal to |b| is:
\[ \mathbf{v} = \frac{7}{3}\hat{i} + \frac{14}{3}\hat{j} + \frac{14}{3}\hat{k} \]

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