Two vectors `vecA=4hati+alphahatj+2hatk" and "vecB=2hati+hatj+hatk` are parallel if :-

Options

A. `alpha=0`
B. `alpha=1`
C. `alpha=2`
D. `alpha=4`

Answer

To determine the value of \( \alpha \) for which the vectors \( \vec{A} = 4\hat{i} + \alpha\hat{j} + 2\hat{k} \) and \( \vec{B} = 2\hat{i} + \hat{j} + \hat{k} \) are parallel, we follow these steps:

### Step 1: Understand the condition for parallel vectors
Two vectors \( \vec{A} \) and \( \vec{B} \) are parallel if their corresponding components are proportional. This means:
\[ \frac{A_x}{B_x} = \frac{A_y}{B_y} = \frac{A_z}{B_z} \]
where \( A_x, A_y, A_z \) are the components of \( \vec{A} \) and \( B_x, B_y, B_z \) are the components of \( \vec{B} \).

### Step 2: Identify the components of the vectors
For the given vectors:
- \( \vec{A} = 4\hat{i} + \alpha\hat{j} + 2\hat{k} \) has components \( A_x = 4 \), \( A_y = \alpha \), \( A_z = 2 \).
- \( \vec{B} = 2\hat{i} + 1\hat{j} + 1\hat{k} \) has components \( B_x = 2 \), \( B_y = 1 \), \( B_z = 1 \).

### Step 3: Set up the proportionality equations
From the condition of proportionality, we have:
\[ \frac{4}{2} = \frac{\alpha}{1} = \frac{2}{1} \]

### Step 4: Solve the first proportion
From the first proportion \( \frac{4}{2} \):
\[ \frac{4}{2} = 2 \]

### Step 5: Solve the second proportion
Now, using the second proportion \( \frac{\alpha}{1} = 2 \):
\[ \alpha = 2 \]

### Step 6: Solve the third proportion
Finally, using the third proportion \( \frac{2}{1} \):
\[ \frac{2}{1} = 2 \]

### Conclusion
Thus, for the vectors \( \vec{A} \) and \( \vec{B} \) to be parallel, the value of \( \alpha \) must be:
\[ \alpha = 2 \]

### Final Answer
The vectors \( \vec{A} \) and \( \vec{B} \) are parallel if \( \alpha = 2 \).

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