If vector `P, Q and R` have magnitude 5,12,and 13 units and `vec(P)+vec(Q)=vec(R )`, the angle between Q and R is

Options

A. `cos^(-1)5/(12)`
B. `cos^(-1)5/(13)`
C. `cos^(-1)(12)/(13)`
D. `cos^(-1)7/(13)`

Explanation

To solve the problem, we need to determine the angle between vectors Q and R given that the magnitudes of vectors P, Q, and R are 5, 12, and 13 units respectively, and that \(\vec{P} + \vec{Q} = \vec{R}\).

### Step-by-Step Solution:

1. **Identify the Vectors and Their Magnitudes**:
- Let \(|\vec{P}| = 5\)
- Let \(|\vec{Q}| = 12\)
- Let \(|\vec{R}| = 13\)

2. **Use the Triangle Law of Vectors**:
- Since \(\vec{P} + \vec{Q} = \vec{R}\), we can visualize this as a triangle where \(\vec{R}\) is the resultant of \(\vec{P}\) and \(\vec{Q}\).

3. **Check for Right Triangle**:
- We can check if the triangle formed by the vectors is a right triangle using the Pythagorean theorem:
\[ |\vec{R}|^2 = |\vec{P}|^2 + |\vec{Q}|^2 \]
- Calculate:
\[ 13^2 = 5^2 + 12^2 \implies 169 = 25 + 144 \implies 169 = 169 \]
- This confirms that the triangle is a right triangle.

4. **Determine the Angle Between Q and R**:
- In a right triangle, we can use trigonometric ratios to find the angle \(\theta\) between \(\vec{Q}\) and \(\vec{R}\).
- Here, \(\vec{Q}\) is adjacent to the angle \(\theta\) and \(\vec{R}\) is the hypotenuse.
- Therefore, we can use the cosine function:
\[ \cos(\theta) = \frac{|\vec{Q}|}{|\vec{R}|} = \frac{12}{13} \]

5. **Calculate the Angle**:
- To find \(\theta\), take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{12}{13}\right) \]

6. **Conclusion**:
- The angle \(\theta\) between vectors Q and R is \(\cos^{-1}\left(\frac{12}{13}\right)\).

### Final Answer:
The angle between vectors Q and R is \(\cos^{-1}\left(\frac{12}{13}\right)\).