Suppose `vecx=(-3,-1), vecy=(-1,4).` Find the magnitude of `vecx+vecy`.

Options

A. 2
B. 3
C. 4
D. 5

Explanation

To find the magnitude of the vector sum \(\vec{x} + \vec{y}\), we will follow these steps:

### Step 1: Represent the vectors
Given:
\[ \vec{x} = (-3, -1) \quad \text{and} \quad \vec{y} = (-1, 4) \]
We can represent these vectors in the form of unit vectors:
\[ \vec{x} = -3 \hat{i} - 1 \hat{j} \quad \text{and} \quad \vec{y} = -1 \hat{i} + 4 \hat{j} \]

### Step 2: Add the vectors
To find \(\vec{x} + \vec{y}\), we add the corresponding components:
\[ \vec{x} + \vec{y} = (-3 \hat{i} - 1 \hat{j}) + (-1 \hat{i} + 4 \hat{j}) \]
Calculating the \(i\) and \(j\) components separately:
\[ \text{For } \hat{i}: -3 - 1 = -4 \]
\[ \text{For } \hat{j}: -1 + 4 = 3 \]
Thus, we have:
\[ \vec{x} + \vec{y} = -4 \hat{i} + 3 \hat{j} \]

### Step 3: Calculate the magnitude
The magnitude of a vector \(\vec{v} = a \hat{i} + b \hat{j}\) is given by:
\[ |\vec{v}| = \sqrt{a^2 + b^2} \]
For our vector \(\vec{x} + \vec{y} = -4 \hat{i} + 3 \hat{j}\):
\[ |\vec{x} + \vec{y}| = \sqrt{(-4)^2 + (3)^2} \]
Calculating the squares:
\[ = \sqrt{16 + 9} = \sqrt{25} \]
Finally, taking the square root:
\[ = 5 \]

### Final Answer
The magnitude of \(\vec{x} + \vec{y}\) is \(5\).

---