The vectors `vec(a)` and `vec(b)` are perpendicular if :

Options

A. `vec(a).vec(b)=0`
B. `vec(a).vec(b)ne 0`
C. `vec(a)xx vec(b)=vec(0)`
D. `vec(a)xx vec(b)ne vec(0)`

Answer

To determine the condition under which the vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular, we can follow these steps:

### Step 1: Understanding Perpendicular Vectors
Vectors \(\vec{a}\) and \(\vec{b}\) are said to be perpendicular if the angle \(\theta\) between them is \(90^\circ\).

### Step 2: Using the Dot Product
The dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is given by the formula:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \]
Where:
- \(|\vec{a}|\) is the magnitude of vector \(\vec{a}\),
- \(|\vec{b}|\) is the magnitude of vector \(\vec{b}\),
- \(\theta\) is the angle between the two vectors.

### Step 3: Substituting the Angle
Since \(\theta = 90^\circ\), we know that:
\[ \cos(90^\circ) = 0 \]
Thus, substituting this into the dot product formula gives:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cdot 0 = 0 \]

### Step 4: Conclusion
From the above steps, we conclude that:
\[ \vec{a} \cdot \vec{b} = 0 \]
This means that the vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular if and only if their dot product is equal to zero.

### Final Answer
The vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular if:
\[ \vec{a} \cdot \vec{b} = 0 \]

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