Three vectors `vecA,vecB` and `vecC` are such that `vecA=vecB+vecC` and their magnitudes are in ratio 5:4:3 respectively. Find angle between vector `vecA` and `vecC`

Explanation

To find the angle between vectors \(\vec{A}\) and \(\vec{C}\) given that \(\vec{A} = \vec{B} + \vec{C}\) and their magnitudes are in the ratio \(5:4:3\), we can follow these steps:

### Step 1: Assign Magnitudes
Let the magnitudes of the vectors be:
- \(|\vec{A}| = 5x\)
- \(|\vec{B}| = 4x\)
- \(|\vec{C}| = 3x\)

### Step 2: Use the Vector Equation
From the equation \(\vec{A} = \vec{B} + \vec{C}\), we can rearrange it to:
\[ \vec{A} - \vec{C} = \vec{B} \]

### Step 3: Apply the Law of Cosines
Taking magnitudes on both sides, we have:
\[ |\vec{A} - \vec{C}| = |\vec{B}| \]
Using the law of cosines for the left side:
\[ |\vec{A} - \vec{C}|^2 = |\vec{A}|^2 + |\vec{C}|^2 - 2 |\vec{A}| |\vec{C}| \cos \theta \]
Substituting the magnitudes:
\[ |\vec{B}|^2 = (5x)^2 + (3x)^2 - 2(5x)(3x)\cos \theta \]

### Step 4: Substitute the Magnitudes
Now substituting the values:
\[ (4x)^2 = (5x)^2 + (3x)^2 - 2(5x)(3x)\cos \theta \]
This simplifies to:
\[ 16x^2 = 25x^2 + 9x^2 - 30x^2 \cos \theta \]

### Step 5: Simplify the Equation
Combine like terms:
\[ 16x^2 = 34x^2 - 30x^2 \cos \theta \]
Rearranging gives:
\[ 30x^2 \cos \theta = 34x^2 - 16x^2 \]
\[ 30x^2 \cos \theta = 18x^2 \]

### Step 6: Solve for \(\cos \theta\)
Dividing both sides by \(30x^2\):
\[ \cos \theta = \frac{18}{30} = \frac{3}{5} \]

### Step 7: Find the Angle \(\theta\)
Now, we can find \(\theta\):
\[ \theta = \cos^{-1}\left(\frac{3}{5}\right) \]

### Final Answer
Thus, the angle between vectors \(\vec{A}\) and \(\vec{C}\) is:
\[ \theta = \cos^{-1}\left(\frac{3}{5}\right) \]

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