Let `vec(a).vec(b) and vec(c )` be the position vectors of points A,B and C, respectively. Under which one of the following conditions are the points A, B and C collinear?

Options

A. A. `vec(a) xx vec(b)= 0`
B. B. `vec(b) xx vec(c )` is parallel to `vec(a) xx vec(b)`
C. C. `vec(a) xx vec(b)` is perpendicular to `vec(b) xx vec(c )`
D. D. `(vec(a) xx vec(b)) + (vec(b) xx vec(c )) + (vec(c ) xx vec(a)) = 0`

Answer

To determine the condition under which the points A, B, and C are collinear, we can use the concept of vectors. Here’s a step-by-step solution:

### Step 1: Define the Position Vectors
Let the position vectors of points A, B, and C be represented as:
- \(\vec{A} = \vec{a}\)
- \(\vec{B} = \vec{b}\)
- \(\vec{C} = \vec{c}\)

### Step 2: Find the Vectors AB and BC
The vector from point A to point B, denoted as \(\vec{AB}\), can be calculated as:
\[ \vec{AB} = \vec{B} - \vec{A} = \vec{b} - \vec{a} \]
Similarly, the vector from point B to point C, denoted as \(\vec{BC}\), is:
\[ \vec{BC} = \vec{C} - \vec{B} = \vec{c} - \vec{b} \]

### Step 3: Use the Cross Product Condition
For points A, B, and C to be collinear, the cross product of vectors \(\vec{AB}\) and \(\vec{BC}\) must equal zero:
\[ \vec{AB} \times \vec{BC} = 0 \]
This implies that the vectors are parallel, which is a condition for collinearity.

### Step 4: Substitute the Vectors
Substituting the expressions for \(\vec{AB}\) and \(\vec{BC}\):
\[ (\vec{b} - \vec{a}) \times (\vec{c} - \vec{b}) = 0 \]

### Step 5: Expand the Cross Product
Expanding this expression using the distributive property of the cross product:
\[ \vec{b} \times \vec{c} - \vec{b} \times \vec{b} - \vec{a} \times \vec{c} + \vec{a} \times \vec{b} = 0 \]
Since \(\vec{b} \times \vec{b} = 0\) (the cross product of any vector with itself is zero), we can simplify this to:
\[ \vec{b} \times \vec{c} - \vec{a} \times \vec{c} + \vec{a} \times \vec{b} = 0 \]

### Step 6: Rearranging the Equation
Rearranging the above equation gives:
\[ \vec{b} \times \vec{c} + \vec{a} \times \vec{b} - \vec{a} \times \vec{c} = 0 \]

### Step 7: Final Condition
Thus, the final condition for the points A, B, and C to be collinear can be expressed as:
\[ \vec{b} \times \vec{c} + \vec{c} \times \vec{a} + \vec{a} \times \vec{b} = 0 \]

### Conclusion
Therefore, the points A, B, and C are collinear if:
\[ \vec{b} \times \vec{c} + \vec{c} \times \vec{a} + \vec{a} \times \vec{b} = 0 \]