OF the vectors given below, the parallel vectors are :
`vec(A) = 6hat(i) + 8hat(j) " "vec(B) = 210 hat(i) + 280hat(k)`
`vec(C ) = 0.3 hat(i) + 0.4 hat(j) " "vec(D) = 3.6 hat(i) + 6hat(j) + 4.8 hat(k)`

Options

A. `vec(A)` and `vec(B)`
B. `vec(A)` and `vec(C )`
C. `vec(A)` and `vec(D)`
D. `vec(C )` and `vec(D)`

Answer

To determine which of the given vectors are parallel, we need to analyze the direction ratios of each vector. Two vectors are parallel if their direction ratios are proportional.

### Given Vectors:
1. \( \vec{A} = 6 \hat{i} + 8 \hat{j} \)
2. \( \vec{B} = 210 \hat{i} + 280 \hat{k} \)
3. \( \vec{C} = 0.3 \hat{i} + 0.4 \hat{j} \)
4. \( \vec{D} = 3.6 \hat{i} + 6 \hat{j} + 4.8 \hat{k} \)

### Step 1: Identify the direction ratios
- For \( \vec{A} \): The direction ratios are \( (6, 8) \).
- For \( \vec{B} \): The direction ratios are \( (210, 0, 280) \).
- For \( \vec{C} \): The direction ratios are \( (0.3, 0.4) \).
- For \( \vec{D} \): The direction ratios are \( (3.6, 6, 4.8) \).

### Step 2: Check for parallelism between \( \vec{A} \) and \( \vec{C} \)
To check if \( \vec{A} \) and \( \vec{C} \) are parallel, we need to see if the ratios of their direction ratios are equal.

- The ratio for \( \vec{A} \) is \( \frac{6}{0.3} = 20 \) and \( \frac{8}{0.4} = 20 \).

Since both ratios are equal, \( \vec{A} \) and \( \vec{C} \) are parallel.

### Step 3: Check for parallelism between \( \vec{B} \) and \( \vec{D} \)
Next, we check \( \vec{B} \) and \( \vec{D} \).

- \( \vec{B} \) has direction ratios \( (210, 0, 280) \).
- \( \vec{D} \) has direction ratios \( (3.6, 6, 4.8) \).

Since \( \vec{B} \) has a non-zero component in the \( \hat{k} \) direction and \( \vec{D} \) has components in all three directions, they cannot be parallel.

### Step 4: Conclude the results
Thus, the only parallel vectors among the given options are \( \vec{A} \) and \( \vec{C} \).

### Final Answer:
The parallel vectors are \( \vec{A} \) and \( \vec{C} \).

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