Which of the following pairs of vectors are parallel ?

Options

A. `vecA = hati - 2 hatj, vecB = hati - 5hatj`
B. `vecA = hati - 10 hatj, vecB = 2hati - 5hatj`
C. `vecA = hati - 5hatj, vecB = hati - 10 hatj`
D. `vecA = hati - 5hatj, vecB = 2hati - 10hatj`

Explanation

To determine which pairs of vectors are parallel, we can use the property of the cross product. Two vectors are parallel if their cross product is zero. Let's denote the vectors as \( \vec{A} \) and \( \vec{B} \). The cross product is given by:

\[ \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin(\theta) \hat{n} \]

Where \( \theta \) is the angle between the two vectors. If \( \theta = 0^\circ \) (meaning the vectors are parallel), then \( \sin(0) = 0 \), and thus \( \vec{A} \times \vec{B} = 0 \).

### Step-by-Step Solution:

1. **Identify the Vectors**: List the pairs of vectors given in the question. For example, let’s say we have:
- Pair 1: \( \vec{A_1} = a_1 \hat{i} + b_1 \hat{j} \)
- Pair 2: \( \vec{A_2} = a_2 \hat{i} + b_2 \hat{j} \)
- Pair 3: \( \vec{A_3} = a_3 \hat{i} + b_3 \hat{j} \)
- Pair 4: \( \vec{A_4} = a_4 \hat{i} + b_4 \hat{j} \)

2. **Calculate the Cross Product**: For each pair of vectors, calculate the cross product. The cross product in two dimensions can be simplified as follows:
\[ \vec{A} \times \vec{B} = (a_1 \hat{i} + b_1 \hat{j}) \times (a_2 \hat{i} + b_2 \hat{j}) = (a_1 b_2 - a_2 b_1) \hat{k} \]

3. **Check for Zero**: For each pair, check if the result of the cross product is zero:
- If \( a_1 b_2 - a_2 b_1 = 0 \), then the vectors are parallel.
- If not, they are not parallel.

4. **Identify Parallel Pairs**: After calculating the cross products for all pairs, identify which pairs resulted in a zero cross product.

5. **Conclusion**: State which pairs of vectors are parallel based on the calculations.

### Example:
Let’s say we have the following pairs:
- Pair 1: \( \vec{A_1} = \hat{i} - 5\hat{j} \)
- Pair 2: \( \vec{A_2} = 2\hat{i} - 10\hat{j} \)

Calculating the cross product for Pair 1 and Pair 2:
\[ \vec{A_1} \times \vec{A_2} = (1)(-10) - (2)(-5) = -10 + 10 = 0 \]

Since the cross product is zero, \( \vec{A_1} \) and \( \vec{A_2} \) are parallel.