A, B, C and D have position vectors `veca, vecb, vecc and vecd`, repectively, such that `veca-vecb = 2(vecd-vecc)`. Then

Options

A. AB and CD bisect each other
B. BD and AC bisect each other
C. AB and CD trisect each other
D. BD and AC trisect each other

Explanation

To solve the problem, we start with the given equation involving position vectors:

### Step 1: Write down the given equation
The given equation is:
\[ \vec{a} - \vec{b} = 2(\vec{d} - \vec{c}) \]

### Step 2: Rearrange the equation
We can rearrange the equation to isolate terms involving \(\vec{a}\) and \(\vec{b}\) on one side, and terms involving \(\vec{c}\) and \(\vec{d}\) on the other side:
\[ \vec{a} - \vec{b} = 2\vec{d} - 2\vec{c} \]

### Step 3: Move \(\vec{c}\) and \(\vec{b}\) to the opposite sides
Now, we can rearrange this equation further:
\[ \vec{a} + 2\vec{c} = \vec{b} + 2\vec{d} \]

### Step 4: Interpret the equation
The equation \(\vec{a} + 2\vec{c} = \vec{b} + 2\vec{d}\) suggests that the vector \(\vec{a}\) and \(\vec{b}\) can be expressed in terms of the vectors \(\vec{c}\) and \(\vec{d}\).

### Step 5: Identify the relationship
This equation indicates that the points represented by the vectors \(\vec{a}, \vec{b}, \vec{c}, \vec{d}\) are such that the segments \(\vec{a}\) to \(\vec{c}\) and \(\vec{b}\) to \(\vec{d}\) trisect each other.

### Final Conclusion
Thus, we conclude that:
\[ \text{The points A, C and B, D trisect each other.} \]

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