If `vec a * vec b = vec a * vec c` then

Options

A. `vec a=0`
B. `vec b =vec c `
C. `vec a _|_ (vec b - vec c )`
D. all of these

Answer

To solve the problem where \( \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} \), we can follow these steps:

### Step 1: Write the equation
We start with the given equation:
\[ \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} \]

### Step 2: Rearrange the equation
We can rearrange this equation to isolate one side:
\[ \vec{a} \cdot \vec{b} - \vec{a} \cdot \vec{c} = 0 \]

### Step 3: Factor out \( \vec{a} \)
Using the distributive property of the dot product, we can factor out \( \vec{a} \):
\[ \vec{a} \cdot (\vec{b} - \vec{c}) = 0 \]

### Step 4: Analyze the result
The equation \( \vec{a} \cdot (\vec{b} - \vec{c}) = 0 \) implies that either:
1. \( \vec{a} = \vec{0} \) (the zero vector), or
2. \( \vec{b} - \vec{c} \) is perpendicular to \( \vec{a} \), which means \( \vec{b} \) is equal to \( \vec{c} \) or \( \vec{a} \) is perpendicular to the vector \( \vec{b} - \vec{c} \).

### Step 5: Conclusion
Thus, we conclude that:
- \( \vec{a} = \vec{0} \) (the zero vector),
- \( \vec{b} = \vec{c} \), or
- \( \vec{a} \) is perpendicular to \( \vec{b} - \vec{c} \).

Therefore, all of these conditions are valid, and the correct answer is that all options are correct.