Which of the following pairs of vectors are parallel ?

Options

A. `vec(A) = hat(i) - 2hat(j) , vec(B) = hat(i) - 5hat(j)`
B. `vec(A) = hat(i) - 10hat(j) , vec(B) = hat(i) - 5hat(j)`
C. `vec(A) = hat(i) - 5hat(j) , vec(B) = hat(i) - 10hat(j)`
D. `vec(A) = hat(i) - 5hat(j) , vec(B) = 2 hat(i) - 10hat(j)`

Answer

To determine which pairs of vectors are parallel, we need to check if their direction ratios are proportional. Two vectors \( \mathbf{A} = a_1 \hat{i} + b_1 \hat{j} \) and \( \mathbf{B} = a_2 \hat{i} + b_2 \hat{j} \) are parallel if the following condition holds:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \]

Let's analyze each pair of vectors step by step.

### Step 1: Analyze the First Pair of Vectors
**Vectors:** \( \mathbf{A_1} = \hat{i} - 2\hat{j} \) and \( \mathbf{B_1} = \hat{i} - 5\hat{j} \)

**Direction Ratios:**
- For \( \mathbf{A_1} \): \( (1, -2) \)
- For \( \mathbf{B_1} \): \( (1, -5) \)

**Check Proportionality:**
\[ \frac{1}{1} \quad \text{and} \quad \frac{-2}{-5} \]
Calculating:
\[ 1 \neq \frac{2}{5} \]
**Conclusion:** These vectors are not parallel.

### Step 2: Analyze the Second Pair of Vectors
**Vectors:** \( \mathbf{A_2} = \hat{i} - \hat{j} \) and \( \mathbf{B_2} = \hat{i} + 5\hat{j} \)

**Direction Ratios:**
- For \( \mathbf{A_2} \): \( (1, -1) \)
- For \( \mathbf{B_2} \): \( (1, 5) \)

**Check Proportionality:**
\[ \frac{1}{1} \quad \text{and} \quad \frac{-1}{5} \]
Calculating:
\[ 1 \neq -\frac{1}{5} \]
**Conclusion:** These vectors are not parallel.

### Step 3: Analyze the Third Pair of Vectors
**Vectors:** \( \mathbf{A_3} = \hat{i} - 5\hat{j} \) and \( \mathbf{B_3} = \hat{i} - 10\hat{j} \)

**Direction Ratios:**
- For \( \mathbf{A_3} \): \( (1, -5) \)
- For \( \mathbf{B_3} \): \( (1, -10) \)

**Check Proportionality:**
\[ \frac{1}{1} \quad \text{and} \quad \frac{-5}{-10} \]
Calculating:
\[ 1 \neq \frac{5}{10} \quad \text{(which simplifies to } \frac{1}{2}\text{)} \]
**Conclusion:** These vectors are not parallel.

### Step 4: Analyze the Fourth Pair of Vectors
**Vectors:** \( \mathbf{A_4} = 2\hat{i} - 5\hat{j} \) and \( \mathbf{B_4} = 4\hat{i} - 10\hat{j} \)

**Direction Ratios:**
- For \( \mathbf{A_4} \): \( (2, -5) \)
- For \( \mathbf{B_4} \): \( (4, -10) \)

**Check Proportionality:**
\[ \frac{2}{4} \quad \text{and} \quad \frac{-5}{-10} \]
Calculating:
\[ \frac{1}{2} = \frac{1}{2} \]
**Conclusion:** These vectors are parallel.

### Final Answer
The only pair of vectors that are parallel is the fourth pair \( \mathbf{A_4} \) and \( \mathbf{B_4} \).

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