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

Options

A. `([(veca,vecb,vecc)])/(|vecaxxvecb+veccxxveca+vecbxxvecc|)`
B. `(2[(veca,vecb,vecc)])/(|vecaxxvecb+vecbxxvecc+veccxxveca|)`
C. `[(veca,vecb,vecc)]`
D. none of these

Explanation

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-by-Step Solution:

1. **Identify the Points**: We have three non-collinear points represented by vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\).

2. **Determine the Normal Vector**: The normal vector to the plane formed by these three points can be found using the cross product of two vectors that lie on the plane. We can take the vectors \(\vec{b} - \vec{a}\) and \(\vec{c} - \vec{b}\):
\[ \vec{n} = (\vec{b} - \vec{a}) \times (\vec{c} - \vec{b}) \]

3. **Equation of the Plane**: The equation of the plane can be expressed in the form:
\[ \vec{r} - \vec{a} \cdot \vec{n} = 0 \]
This can be rearranged to:
\[ \vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n} \]

4. **Finding the Length of the Perpendicular**: The length of the perpendicular from the origin (point \(\vec{0}\)) to the plane can be calculated using the formula:
\[ \text{Length} = \frac{|\vec{a} \cdot \vec{n}|}{|\vec{n}|} \]
Here, \(\vec{n}\) is the normal vector we computed in step 2.

5. **Substituting Values**: We can express \(\vec{n}\) in terms of the vectors:
\[ \vec{n} = (\vec{b} - \vec{a}) \times (\vec{c} - \vec{b}) \]
Then, substituting this into the length formula gives:
\[ \text{Length} = \frac{|\vec{a} \cdot ((\vec{b} - \vec{a}) \times (\vec{c} - \vec{b}))|}{|(\vec{b} - \vec{a}) \times (\vec{c} - \vec{b})|} \]

6. **Final Expression**: After simplifying, we arrive at the final expression for the length of the perpendicular from the origin to the plane:
\[ \text{Length} = \frac{|\vec{a} \cdot \vec{n}|}{|\vec{n}|} = \frac{\vec{a} \cdot \vec{b} \times \vec{c}}{|\vec{b} \times \vec{c}|} \]

### Final Result:
The length of the perpendicular from the origin to the plane is given by:
\[ \text{Length} = \frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{|\vec{b} \times \vec{c}|} \]