If `a=(2,5) and b=(1,4)`, then vector parallel to (a+b) is

Options

A. (3,5)
B. (1,1)
C. (1,3)
D. (8,5)

Explanation

To find a vector parallel to \( \mathbf{a} + \mathbf{b} \), we will follow these steps:

### Step 1: Define the vectors
Given:
\[ \mathbf{a} = (2, 5) \quad \text{and} \quad \mathbf{b} = (1, 4) \]
We can express these vectors in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \):
\[ \mathbf{a} = 2\mathbf{i} + 5\mathbf{j} \]
\[ \mathbf{b} = 1\mathbf{i} + 4\mathbf{j} \]

### Step 2: Add the vectors
Now, we will add the two vectors \( \mathbf{a} \) and \( \mathbf{b} \):
\[ \mathbf{a} + \mathbf{b} = (2\mathbf{i} + 5\mathbf{j}) + (1\mathbf{i} + 4\mathbf{j}) \]
Combine the \( \mathbf{i} \) components and the \( \mathbf{j} \) components:
\[ \mathbf{a} + \mathbf{b} = (2 + 1)\mathbf{i} + (5 + 4)\mathbf{j} = 3\mathbf{i} + 9\mathbf{j} \]

### Step 3: Identify the vector parallel to \( \mathbf{a} + \mathbf{b} \)
The resultant vector \( \mathbf{a} + \mathbf{b} \) is:
\[ \mathbf{a} + \mathbf{b} = 3\mathbf{i} + 9\mathbf{j} \]
A vector parallel to this resultant vector can be expressed as:
\[ k(3\mathbf{i} + 9\mathbf{j}) \quad \text{for any scalar } k \]
For simplicity, we can choose \( k = 1 \):
\[ \text{A vector parallel to } \mathbf{a} + \mathbf{b} \text{ is } (3, 9) \]

### Final Answer
A vector parallel to \( \mathbf{a} + \mathbf{b} \) is \( (3, 9) \).

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