If `overset(rarr)A=overset(rarr)B+overset(rarr)C` and the magnitude of `overset(rarr)A, overset(rarr)B` and `overset(rarr)C` are 5,4 and 3 units respectively the angle between `overset(rarr)A` and `overset(rarr)C` is

Options

A. `cos^(-1)3/5`
B. `cos^(-1)4/5`
C. `(pi)/(2)`
D. `sin^(-1)3/4`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We are given that:
- \(\vec{A} = \vec{B} + \vec{C}\)
- Magnitude of \(\vec{A} = 5\) units
- Magnitude of \(\vec{B} = 4\) units
- Magnitude of \(\vec{C} = 3\) units

### Step 2: Rewrite the equation
We can express \(\vec{B}\) in terms of \(\vec{A}\) and \(\vec{C}\):
\[ \vec{B} = \vec{A} - \vec{C} \]

### Step 3: Apply the law of cosines
The magnitude of \(\vec{B}\) can be expressed using the law of cosines:
\[ |\vec{B}|^2 = |\vec{A}|^2 + |\vec{C}|^2 - 2 |\vec{A}| |\vec{C}| \cos(\theta) \]
where \(\theta\) is the angle between \(\vec{A}\) and \(\vec{C}\).

### Step 4: Substitute the known values
Substituting the magnitudes into the equation:
\[ 4^2 = 5^2 + 3^2 - 2 \cdot 5 \cdot 3 \cdot \cos(\theta) \]
This simplifies to:
\[ 16 = 25 + 9 - 30 \cos(\theta) \]

### Step 5: Simplify the equation
Combine the constants:
\[ 16 = 34 - 30 \cos(\theta) \]
Rearranging gives:
\[ 30 \cos(\theta) = 34 - 16 \]
\[ 30 \cos(\theta) = 18 \]

### Step 6: Solve for \(\cos(\theta)\)
Dividing both sides by 30:
\[ \cos(\theta) = \frac{18}{30} = \frac{3}{5} \]

### Step 7: Find the angle \(\theta\)
To find the angle \(\theta\), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{3}{5}\right) \]

### Final Answer
The angle between \(\vec{A}\) and \(\vec{C}\) is \(\cos^{-1}\left(\frac{3}{5}\right)\).

---