To determine which of the given vectors are parallel, we need to check if any of the vectors can be expressed as a scalar multiple of another vector.
### Given Vectors:
1. **Vector A**: \(\vec{A} = 6\hat{i} + 8\hat{j}\)
2. **Vector B**: \(\vec{B} = 210\hat{i} + 280\hat{k}\)
3. **Vector C**: \(\vec{C} = 0.3\hat{i} + 0.4\hat{j}\)
4. **Vector D**: \(\vec{D} = 3.6\hat{i} + 6\hat{j} + 4.8\hat{k}\)
### Step 1: Check if A and B are parallel
To check if \(\vec{A}\) and \(\vec{B}\) are parallel, we can express them in terms of their components:
- \(\vec{A} = (6, 8, 0)\)
- \(\vec{B} = (210, 0, 280)\)
For two vectors to be parallel, the ratios of their corresponding components must be equal:
\[
\frac{6}{210} \neq \frac{8}{0} \quad \text{(undefined)}
\]
Since the second component of \(\vec{B}\) is non-zero while that of \(\vec{A}\) is zero, they are not parallel.
### Step 2: Check if A and C are parallel
Now let's check \(\vec{A}\) and \(\vec{C}\):
- \(\vec{C} = (0.3, 0.4, 0)\)
The ratios of their components:
\[
\frac{6}{0.3} = 20 \quad \text{and} \quad \frac{8}{0.4} = 20
\]
Since both ratios are equal, \(\vec{A}\) and \(\vec{C}\) are parallel.
### Step 3: Check if A and D are parallel
Now let's check \(\vec{A}\) and \(\vec{D}\):
- \(\vec{D} = (3.6, 6, 4.8)\)
The ratios of their components:
\[
\frac{6}{3.6} = \frac{5}{3} \quad \text{and} \quad \frac{8}{6} = \frac{4}{3}
\]
Since the ratios are not equal, \(\vec{A}\) and \(\vec{D}\) are not parallel.
### Step 4: Check if B and C are parallel
Now let's check \(\vec{B}\) and \(\vec{C}\):
- \(\vec{B} = (210, 0, 280)\)
The ratios of their components:
\[
\frac{0.3}{210} \neq \frac{0.4}{0} \quad \text{(undefined)}
\]
Since the second component of \(\vec{B}\) is zero while that of \(\vec{C}\) is non-zero, they are not parallel.
### Step 5: Check if B and D are parallel
Now let's check \(\vec{B}\) and \(\vec{D}\):
- \(\vec{D} = (3.6, 6, 4.8)\)
The ratios of their components:
\[
\frac{210}{3.6} \neq \frac{0}{6} \quad \text{(undefined)}
\]
Since the second component of \(\vec{B}\) is zero while that of \(\vec{D}\) is non-zero, they are not parallel.
### Step 6: Check if C and D are parallel
Finally, let's check \(\vec{C}\) and \(\vec{D}\):
- The ratios of their components:
\[
\frac{0.3}{3.6} \neq \frac{0.4}{6} \quad \text{(not equal)}
\]
Since the ratios are not equal, \(\vec{C}\) and \(\vec{D}\) are not parallel.
### Conclusion
The only pair of parallel vectors among the given vectors is:
- **Vectors A and C**.
