The resultant of the forces `vecF_(1) = 4hati-3hatj` and `vecF_(2) = 6hati + 8hatj` is

Options

A. `5sqrt(5)`
B. `10hati - 5hatj`
C. 125
D. `-2hati - 3hatj`

Answer

To find the resultant of the forces \(\vec{F_1} = 4\hat{i} - 3\hat{j}\) and \(\vec{F_2} = 6\hat{i} + 8\hat{j}\), we will follow these steps:

### Step 1: Write down the given vectors
We have:
\[ \vec{F_1} = 4\hat{i} - 3\hat{j} \]
\[ \vec{F_2} = 6\hat{i} + 8\hat{j} \]

### Step 2: Add the vectors
To find the resultant vector \(\vec{F_R}\), we add \(\vec{F_1}\) and \(\vec{F_2}\):
\[ \vec{F_R} = \vec{F_1} + \vec{F_2} \]

### Step 3: Combine the components
Now, we combine the \(\hat{i}\) components and the \(\hat{j}\) components separately:
\[ \vec{F_R} = (4 + 6)\hat{i} + (-3 + 8)\hat{j} \]

### Step 4: Calculate the components
Calculating the components:
- For \(\hat{i}\):
\[ 4 + 6 = 10 \]
- For \(\hat{j}\):
\[ -3 + 8 = 5 \]

### Step 5: Write the resultant vector
Now we can write the resultant vector:
\[ \vec{F_R} = 10\hat{i} + 5\hat{j} \]

### Step 6: Conclusion
Thus, the resultant of the forces \(\vec{F_1}\) and \(\vec{F_2}\) is:
\[ \vec{F_R} = 10\hat{i} + 5\hat{j} \]

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