Two vectors 3 units and 5 units are acting at `60^(@)` to each other. What is the magnitude and direction of the resultant

Explanation

To find the magnitude and direction of the resultant of two vectors with magnitudes 3 units and 5 units acting at an angle of 60 degrees to each other, we can follow these steps:

### Step 1: Identify the given values
- Magnitude of vector A (|A|) = 3 units
- Magnitude of vector B (|B|) = 5 units
- Angle between the vectors (θ) = 60 degrees

### Step 2: Use the formula for the magnitude of the resultant vector
The magnitude of the resultant vector R can be calculated using the formula:
\[ |R| = \sqrt{|A|^2 + |B|^2 + 2|A||B|\cos(\theta)} \]

### Step 3: Substitute the known values into the formula
\[ |R| = \sqrt{3^2 + 5^2 + 2 \cdot 3 \cdot 5 \cdot \cos(60^\circ)} \]

### Step 4: Calculate the individual components
- Calculate \(3^2 = 9\)
- Calculate \(5^2 = 25\)
- Calculate \(\cos(60^\circ) = \frac{1}{2}\)
- Calculate \(2 \cdot 3 \cdot 5 \cdot \frac{1}{2} = 15\)

### Step 5: Combine the results
\[ |R| = \sqrt{9 + 25 + 15} = \sqrt{49} \]

### Step 6: Find the magnitude of the resultant
\[ |R| = 7 \text{ units} \]

### Step 7: Determine the direction of the resultant vector
To find the direction, we can use the formula for the angle α (the angle of the resultant vector with respect to vector A):
\[ \tan(\alpha) = \frac{|B| \sin(\theta)}{|A| + |B| \cos(\theta)} \]

### Step 8: Substitute the known values into the direction formula
\[ \tan(\alpha) = \frac{5 \cdot \sin(60^\circ)}{3 + 5 \cdot \cos(60^\circ)} \]

### Step 9: Calculate the individual components for direction
- Calculate \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)
- Calculate \(5 \cdot \sin(60^\circ) = 5 \cdot \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2}\)
- Calculate \(5 \cdot \cos(60^\circ) = 5 \cdot \frac{1}{2} = \frac{5}{2}\)
- Combine the denominator: \(3 + \frac{5}{2} = \frac{6}{2} + \frac{5}{2} = \frac{11}{2}\)

### Step 10: Substitute these values into the tangent formula
\[ \tan(\alpha) = \frac{\frac{5\sqrt{3}}{2}}{\frac{11}{2}} = \frac{5\sqrt{3}}{11} \]

### Step 11: Calculate α using the arctan function
\[ \alpha = \tan^{-1}\left(\frac{5\sqrt{3}}{11}\right) \approx 38.2^\circ \]

### Final Result
- Magnitude of the resultant vector, |R| = 7 units
- Direction of the resultant vector, α ≈ 38.2 degrees

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