Let a,b and c be three unit vectors such that `3a+4b+5c=0`. Then which of the following statements is true?

Options

A. a is parallel to b
B. a is perpendicular to b
C. a is neither parallel nor perpendicular to b
D. none of these

Explanation

To solve the problem, we need to analyze the equation given and the properties of the vectors involved.

### Step 1: Understand the given equation
We are given that:
\[ 3\mathbf{a} + 4\mathbf{b} + 5\mathbf{c} = 0 \]
This means that the vector sum of \(3\mathbf{a}\), \(4\mathbf{b}\), and \(5\mathbf{c}\) equals the zero vector.

### Step 2: Rearranging the equation
We can rearrange the equation to isolate one of the vectors:
\[ 3\mathbf{a} + 4\mathbf{b} = -5\mathbf{c} \]

### Step 3: Squaring both sides
To analyze the relationship between the vectors, we can square both sides of the equation:
\[ \|3\mathbf{a} + 4\mathbf{b}\|^2 = \|-5\mathbf{c}\|^2 \]
This gives us:
\[ (3\mathbf{a} + 4\mathbf{b}) \cdot (3\mathbf{a} + 4\mathbf{b}) = 25\|\mathbf{c}\|^2 \]

### Step 4: Expanding the left-hand side
Now, we expand the left-hand side using the properties of dot products:
\[ (3\mathbf{a}) \cdot (3\mathbf{a}) + 2(3\mathbf{a}) \cdot (4\mathbf{b}) + (4\mathbf{b}) \cdot (4\mathbf{b}) = 25 \]
This simplifies to:
\[ 9\|\mathbf{a}\|^2 + 24(\mathbf{a} \cdot \mathbf{b}) + 16\|\mathbf{b}\|^2 = 25 \]

### Step 5: Substituting the magnitudes
Since \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are unit vectors, we have:
\(\|\mathbf{a}\|^2 = 1\) and \(\|\mathbf{b}\|^2 = 1\).
Substituting these values into the equation gives us:
\[ 9(1) + 24(\mathbf{a} \cdot \mathbf{b}) + 16(1) = 25 \]
This simplifies to:
\[ 9 + 24(\mathbf{a} \cdot \mathbf{b}) + 16 = 25 \]
\[ 25 + 24(\mathbf{a} \cdot \mathbf{b}) = 25 \]

### Step 6: Solving for the dot product
Now, we can isolate the dot product term:
\[ 24(\mathbf{a} \cdot \mathbf{b}) = 0 \]
This implies:
\[ \mathbf{a} \cdot \mathbf{b} = 0 \]

### Step 7: Conclusion
Since the dot product of \(\mathbf{a}\) and \(\mathbf{b}\) is zero, this means that the vectors \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular to each other.

### Final Answer
Thus, the correct statement is that **\(\mathbf{a}\) is perpendicular to \(\mathbf{b}\)**.

---