If `vec(a), vec(b) and vec(c )` are the position vectors of the vertices of an equilateral triangle whose orthocenter is at the origin, and then which one of the following is correct?

Options

A. `vec(a) + vec(b) + vec(c ) = vec(0)`
B. `vec(a) + vec(b) + vec(c )`= unit vector
C. `vec(a) + vec(b) = vec(c )`
D. `vec(a) = vec(b) + vec(c )`

Answer

To solve the problem, we need to analyze the properties of the equilateral triangle and its orthocenter. Let's denote the position vectors of the vertices of the equilateral triangle as \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).

### Step 1: Understand the properties of the equilateral triangle
In an equilateral triangle, the centroid, orthocenter, and circumcenter all coincide. This means that if the orthocenter is at the origin, then the centroid is also at the origin.

### Step 2: Calculate the centroid
The centroid \(G\) of a triangle with vertices represented by position vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) is given by:
\[ \vec{G} = \frac{\vec{a} + \vec{b} + \vec{c}}{3} \]

### Step 3: Set the centroid to the origin
Since the orthocenter is at the origin, we have:
\[ \vec{G} = \vec{0} \]
Thus, we can set the equation for the centroid to zero:
\[ \frac{\vec{a} + \vec{b} + \vec{c}}{3} = \vec{0} \]

### Step 4: Solve for the sum of the position vectors
Multiplying both sides of the equation by 3 gives:
\[ \vec{a} + \vec{b} + \vec{c} = \vec{0} \]

### Conclusion
The correct conclusion from the above steps is that the sum of the position vectors of the vertices of the equilateral triangle is zero:
\[ \vec{a} + \vec{b} + \vec{c} = \vec{0} \]