If `vecA=3hati-4hatj and vecB=- hati- 4hatj`, calculate the direction of `vecA+vecB`

Options

A. `tan^(-1)(4)` with +x- axis in clock wise
B. `tan^(-1)(4)` with -x- axis in clock wise
C. `tan^(-1)(4)` with +x- axis in anti clock wise
D. `tan^(-1)(4)` with -x- axis in anti clock wise

Answer

To solve the problem of finding the direction of the vector \(\vec{A} + \vec{B}\), we will follow these steps:

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

### Step 2: Add the vectors
To find \(\vec{A} + \vec{B}\), we will add the corresponding components of \(\vec{A}\) and \(\vec{B}\).

\[ \vec{A} + \vec{B} = (3\hat{i} - 4\hat{j}) + (-\hat{i} - 4\hat{j}) \]

Combine the \(\hat{i}\) components and the \(\hat{j}\) components:
\[ \vec{A} + \vec{B} = (3 - 1)\hat{i} + (-4 - 4)\hat{j} = 2\hat{i} - 8\hat{j} \]

### Step 3: Determine the components of the resultant vector
The resultant vector is:
\[ \vec{R} = 2\hat{i} - 8\hat{j} \]
This means that the x-component is \(2\) and the y-component is \(-8\).

### Step 4: Calculate the direction of the resultant vector
The direction (angle \(\theta\)) of the vector can be calculated using the tangent function:
\[ \tan(\theta) = \frac{y}{x} = \frac{-8}{2} = -4 \]

### Step 5: Find the angle \(\theta\)
To find \(\theta\), we take the arctangent:
\[ \theta = \tan^{-1}(-4) \]

### Step 6: Determine the quadrant
Since the x-component is positive and the y-component is negative, the vector lies in the fourth quadrant. In the fourth quadrant, the angle is measured clockwise from the positive x-axis.

### Step 7: Adjust the angle for the fourth quadrant
The angle in the fourth quadrant can be expressed as:
\[ \theta = 360^\circ + \tan^{-1}(-4) \quad \text{(or simply use the positive acute angle)} \]
However, we typically express the angle as:
\[ \theta = \tan^{-1}(4) \quad \text{(as the acute angle)} \]

### Final Result
Thus, the direction of the vector \(\vec{A} + \vec{B}\) is:
\[ \theta = \tan^{-1}(4) \quad \text{(measured clockwise from the positive x-axis)} \]

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