IF `veca, vecb, vecc` are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then

Options

A. `veca+vecb+vecc=vec0`
B. `veca^(2)=vecb^(2)+vecc^(2)`
C. `veca+vecb=vecc`
D. none of these

Answer

To solve the problem, we need to analyze the properties of an equilateral triangle and the implications of having its orthocenter at the origin.

### Step 1: Understanding the Orthocenter
The orthocenter of a triangle is the point where the three altitudes intersect. For an equilateral triangle, the orthocenter coincides with the centroid and the circumcenter, all of which are located at the same point.

### Step 2: Position Vectors
Let the position vectors of the vertices of the equilateral triangle be represented as:
- \( \vec{a} \) for vertex A
- \( \vec{b} \) for vertex B
- \( \vec{c} \) for vertex C

### Step 3: Condition for the Orthocenter
Since the orthocenter is at the origin, we can express this condition mathematically. The centroid of the triangle, which is the average of the position vectors of the vertices, must also be at the origin.

### Step 4: Setting Up the Equation
The centroid \( G \) of triangle ABC is given by:
\[ \vec{G} = \frac{\vec{a} + \vec{b} + \vec{c}}{3} \]
For the centroid to be at the origin, we set:
\[ \vec{G} = \vec{0} \]
This leads to the equation:
\[ \frac{\vec{a} + \vec{b} + \vec{c}}{3} = \vec{0} \]

### Step 5: Solving for the Position Vectors
Multiplying both sides of the equation by 3 gives:
\[ \vec{a} + \vec{b} + \vec{c} = \vec{0} \]

### Conclusion
Thus, we conclude that the sum of the position vectors of the vertices of the equilateral triangle is zero:
\[ \vec{a} + \vec{b} + \vec{c} = 0 \]

### Final Answer
The correct option is:
1. \( \vec{a} + \vec{b} + \vec{c} = 0 \)

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