Given : `vecA = 2hati + 3hatj` and `vecB = 5hati - 6hatj`. The magnitude of `vecA + vecB` is

Options

A. 4 units
B. 10 units
C. `sqrt(58)` units
D. `sqrt(61)` units

Answer

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

### Step 1: Write down the vectors
Given:
\[ \vec{A} = 2\hat{i} + 3\hat{j} \]
\[ \vec{B} = 5\hat{i} - 6\hat{j} \]

### Step 2: Add the vectors
To find \(\vec{A} + \vec{B}\), we add the corresponding components of the vectors:
\[ \vec{A} + \vec{B} = (2\hat{i} + 3\hat{j}) + (5\hat{i} - 6\hat{j}) \]
Combine the \(\hat{i}\) components and the \(\hat{j}\) components:
\[ \vec{A} + \vec{B} = (2 + 5)\hat{i} + (3 - 6)\hat{j} \]
\[ \vec{A} + \vec{B} = 7\hat{i} - 3\hat{j} \]

### Step 3: Calculate the magnitude of \(\vec{A} + \vec{B}\)
The magnitude of a vector \(\vec{R} = x\hat{i} + y\hat{j}\) is given by:
\[ |\vec{R}| = \sqrt{x^2 + y^2} \]
For \(\vec{A} + \vec{B} = 7\hat{i} - 3\hat{j}\):
\[ |\vec{A} + \vec{B}| = \sqrt{(7)^2 + (-3)^2} \]
Calculating the squares:
\[ |\vec{A} + \vec{B}| = \sqrt{49 + 9} \]
\[ |\vec{A} + \vec{B}| = \sqrt{58} \]

### Final Answer
Thus, the magnitude of \(\vec{A} + \vec{B}\) is:
\[ |\vec{A} + \vec{B}| = \sqrt{58} \]

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