Two forces `vec(F_(1)) and vec(F_(2))` are acting at right angles to each other , find their resultant ?

Options

A. `sqrt(F_(1)^(2)-F_(2)^(2))`
B. `sqrt(F_(1)^(2)+F_(2)^(2))`
C. `sqrt((F_(1)+F_(2))`
D. `F_(1)+F_(2)`

Answer

To find the resultant of two forces \( \vec{F_1} \) and \( \vec{F_2} \) acting at right angles to each other, we can follow these steps:

### Step 1: Understand the Problem
We have two forces \( \vec{F_1} \) and \( \vec{F_2} \) that are acting at an angle of 90 degrees (right angles) to each other.

### Step 2: Use the Formula for Resultant Force
When two vectors are at right angles, the magnitude of the resultant vector \( \vec{R} \) can be calculated using the Pythagorean theorem. The formula is given by:

\[ R = \sqrt{F_1^2 + F_2^2} \]

### Step 3: Substitute the Values
Since the angle \( \theta \) between \( \vec{F_1} \) and \( \vec{F_2} \) is 90 degrees, we can ignore the cosine term in the general formula for the resultant of two vectors:

\[ R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2 \cos(90^\circ)} \]

Since \( \cos(90^\circ) = 0 \), the equation simplifies to:

\[ R = \sqrt{F_1^2 + F_2^2} \]

### Step 4: Conclusion
Thus, the resultant force \( \vec{R} \) acting at right angles to each other is given by:

\[ R = \sqrt{F_1^2 + F_2^2} \]