If `vecA=4hati-3hatj` and `vecB=6hati+8hatj` then magnitude and direction of `vecA+vecB` will be

Options

A. `5, tan^(-1)(3//4)`
B. `5sqrt(5),tan^(-1)(1//2)`
C. `10,tan^(-1)(5)`
D. `25, tan^(-1)(3//4)`

Answer

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

### Step 1: Write down the vectors
Given:
\[ \vec{A} = 4\hat{i} - 3\hat{j} \]
\[ \vec{B} = 6\hat{i} + 8\hat{j} \]

### Step 2: Add the vectors
To find \(\vec{A} + \vec{B}\), we add the components of \(\vec{A}\) and \(\vec{B}\):
\[ \vec{A} + \vec{B} = (4\hat{i} - 3\hat{j}) + (6\hat{i} + 8\hat{j}) \]
Combine the \(\hat{i}\) components and the \(\hat{j}\) components:
\[ \vec{A} + \vec{B} = (4 + 6)\hat{i} + (-3 + 8)\hat{j} \]
\[ \vec{A} + \vec{B} = 10\hat{i} + 5\hat{j} \]

### Step 3: Calculate the magnitude of \(\vec{A} + \vec{B}\)
The magnitude of a vector \(\vec{C} = x\hat{i} + y\hat{j}\) is given by:
\[ |\vec{C}| = \sqrt{x^2 + y^2} \]
For \(\vec{A} + \vec{B} = 10\hat{i} + 5\hat{j}\):
\[ |\vec{A} + \vec{B}| = \sqrt{(10)^2 + (5)^2} \]
\[ |\vec{A} + \vec{B}| = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5} \]

### Step 4: Calculate the direction of \(\vec{A} + \vec{B}\)
The direction (angle \(\theta\)) of the vector can be found using the tangent function:
\[ \tan(\theta) = \frac{y}{x} \]
For \(\vec{A} + \vec{B} = 10\hat{i} + 5\hat{j}\):
\[ \tan(\theta) = \frac{5}{10} = \frac{1}{2} \]
To find \(\theta\):
\[ \theta = \tan^{-1}\left(\frac{1}{2}\right) \]

### Final Answer
- The magnitude of \(\vec{A} + \vec{B}\) is \(5\sqrt{5}\).
- The direction of \(\vec{A} + \vec{B}\) is \(\tan^{-1}\left(\frac{1}{2}\right)\).