Understanding the Problem
* Resultant Vector: The resultant vector is the vector sum of two or more vectors. It represents the combined effect of the individual vectors.
* Magnitude: The magnitude of a vector is its length or size.
Formula and Explanation
The magnitude of the resultant vector (R) for two vectors (A and B) at an angle (θ) can be calculated using the following formula:
```
R = √(A² + B² + 2AB cos θ)
```
Explanation:
* A² + B²: This part represents the sum of the squares of the magnitudes of the individual vectors.
* 2AB cos θ: This part accounts for the contribution of the angle between the vectors. The cosine of the angle is positive for angles less than 90 degrees, indicating that the vectors contribute constructively to the resultant.
Applying the Formula
Since you have an angle of 60 degrees, we can plug it into the formula:
```
R = √(A² + B² + 2AB cos 60°)
```
Remember that cos 60° = 1/2. So, the formula simplifies to:
```
R = √(A² + B² + AB)
```
Example
Let's say vector A has a magnitude of 5 units and vector B has a magnitude of 3 units. The magnitude of the resultant vector would be:
```
R = √(5² + 3² + 5*3) = √(25 + 9 + 15) = √49 = 7 units
```
Key Points
* The magnitude of the resultant vector is always greater than or equal to the difference of the magnitudes of the individual vectors and less than or equal to the sum of the magnitudes of the individual vectors.
* When the angle between the vectors is 0 degrees (parallel vectors), the resultant vector has the maximum magnitude, which is the sum of the individual vectors.
* When the angle between the vectors is 180 degrees (anti-parallel vectors), the resultant vector has the minimum magnitude, which is the difference of the individual vectors.
Let me know if you have any other vector problems you'd like help with!
