If ` vec(AB) = vecb and vec(AC) =vecc` then the length of the perpendicular from A to the line BC is

Options

A. `(|vecb xx vecc|)/(|vecb + vecc|)`
B. ` (|vecb xx vecc|)/(| vecb -vecc|) `
C. ` (| vecb xx vecc|)/(2 |vecb -vecc|)`
D. ` (|vecb xx vecc|)/( 2|vecb + vecc|) `

Explanation

To find the length of the perpendicular from point A to the line BC given that \( \vec{AB} = \vec{b} \) and \( \vec{AC} = \vec{c} \), we can follow these steps:

### Step 1: Understand the Geometry
We have a triangle ABC where:
- Point A is the vertex from which we are dropping a perpendicular to line BC.
- Vectors \( \vec{AB} \) and \( \vec{AC} \) are represented by \( \vec{b} \) and \( \vec{c} \) respectively.

### Step 2: Area of Triangle ABC
The area of triangle ABC can be expressed in two ways:
1. Using base BC and height from A to BC:
\[ \text{Area} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times |\vec{BC}| \times h \]
where \( h \) is the height from A to line BC.

2. Using the cross product of vectors \( \vec{AB} \) and \( \vec{AC} \):
\[ \text{Area} = \frac{1}{2} |\vec{AB} \times \vec{AC}| = \frac{1}{2} |\vec{b} \times \vec{c}| \]

### Step 3: Equate the Two Area Expressions
Since both expressions represent the same area, we can set them equal to each other:
\[ \frac{1}{2} |\vec{BC}| \times h = \frac{1}{2} |\vec{b} \times \vec{c}| \]

### Step 4: Solve for Height \( h \)
Canceling \( \frac{1}{2} \) from both sides gives:
\[ |\vec{BC}| \times h = |\vec{b} \times \vec{c}| \]
Now, we can isolate \( h \):
\[ h = \frac{|\vec{b} \times \vec{c}|}{|\vec{BC}|} \]

### Step 5: Find \( |\vec{BC}| \)
The vector \( \vec{BC} \) can be found as:
\[ \vec{BC} = \vec{C} - \vec{B} \]
Thus, the magnitude of \( \vec{BC} \) is:
\[ |\vec{BC}| = |\vec{C} - \vec{B}| \]

### Final Expression for Height \( h \)
Substituting this back into our equation for \( h \):
\[ h = \frac{|\vec{b} \times \vec{c}|}{|\vec{C} - \vec{B}|} \]

### Conclusion
The length of the perpendicular from point A to the line BC is given by:
\[ h = \frac{|\vec{b} \times \vec{c}|}{|\vec{C} - \vec{B}|} \]