The length of the perpendicular from the origin to the plane passing though three non-collinear points `veca,vecb,vecc` is

Options

A. `([[vecavecbvecc]])/([vecaxxvecb+ vecbxx vecc+ vecc xx veca])`
B. `(2[[vecavecbvecc]])/([vecaxxvecb+ vecbxx vecc+ vecc xx veca])`
C. `[vecavecbvecc]`
D. None of these

Answer

To find the length of the perpendicular from the origin to the plane passing through three non-collinear points represented by vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), we can follow these steps:

### Step 1: Understand the Plane Equation
The equation of the plane passing through the points represented by vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) can be derived from the vector equation of a plane. The general form of the equation of a plane through three points is given by:
\[ \vec{R} \cdot (\vec{a} \times \vec{b}) + \vec{c} \cdot (\vec{a} \times \vec{b}) = 0 \]
This indicates that the normal vector to the plane is \(\vec{n} = \vec{a} \times \vec{b}\).

### Step 2: Find the Normal Vector
To find the normal vector to the plane, we can use the cross products of the vectors:
\[ \vec{n} = \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} \]
This normal vector is essential for calculating the perpendicular distance from the origin to the plane.

### Step 3: Calculate the Distance from the Origin
The length of the perpendicular \(l\) from the origin to the plane can be calculated using the formula:
\[ l = \frac{|\vec{a} \cdot \vec{n}|}{|\vec{n}|} \]
Where \(\vec{n}\) is the normal vector calculated in the previous step.

### Step 4: Substitute the Normal Vector
Substituting the expression for \(\vec{n}\):
\[ l = \frac{|\vec{a} \cdot (\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a})|}{|\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|} \]
This gives us the length of the perpendicular from the origin to the plane.

### Step 5: Simplify the Expression
After simplification, we can express the length of the perpendicular as:
\[ l = \frac{|\vec{a} \cdot \vec{b} \cdot \vec{c}|}{|\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|} \]

### Conclusion
After evaluating the options provided in the question, we find that the expression derived matches with Option 1. Therefore, the length of the perpendicular from the origin to the plane passing through the three non-collinear points is given by:
\[ l = \frac{|\vec{a} \cdot \vec{b} \cdot \vec{c}|}{|\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|} \]

### Final Answer
Thus, the correct option is **Option 1**.

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