The perpendicular distance of the point `vecc` from the joining ` veca and vecb` is

Options

A. `(|vecbxx vecc +vecc xx veca +veca xx vecb|)/(|vecb -veca|)`
B. `(|vecaxx vecb +vecb xx vecc +vecc xx veca|)/(|vecb -veca|)`
C. ` (|veca xx vecb + vecb xx vecc + vecc xx veca|)/( |veca - veca|)`
D. ` 1/2 (|veca xx vecb + vecb xx vecc + vecc xx veca|)/ (|vecb -veca|) `

Explanation

To find the perpendicular distance of the point \( \vec{C} \) from the line joining \( \vec{A} \) and \( \vec{B} \), we can follow these steps:

### Step 1: Define the Position Vectors
Let \( \vec{A}, \vec{B}, \vec{C} \) be the position vectors of points A, B, and C respectively.

### Step 2: Identify the Line Segment
The line segment joining points A and B can be represented by the vector \( \vec{AB} = \vec{B} - \vec{A} \).

### Step 3: Determine the Perpendicular from C to AB
Let \( M \) be the foot of the perpendicular dropped from point C to the line AB. The length of this perpendicular, \( CM \), represents the distance we want to find.

### Step 4: Use the Area of Triangle Formula
The area of triangle ABC can be expressed in terms of the base \( AB \) and height \( CM \):
\[ \text{Area} = \frac{1}{2} \times |\vec{AB}| \times |CM| \]
We can also express the area using the cross product:
\[ \text{Area} = \frac{1}{2} |\vec{A} \times \vec{B}| \]

### Step 5: Relate the Two Area Expressions
Setting the two area expressions equal gives:
\[ \frac{1}{2} |\vec{AB}| \times |CM| = \frac{1}{2} |\vec{A} \times \vec{B}| \]
Cancelling \( \frac{1}{2} \) from both sides, we have:
\[ |\vec{AB}| \times |CM| = |\vec{A} \times \vec{B}| \]

### Step 6: Solve for CM
Rearranging the equation to solve for \( |CM| \):
\[ |CM| = \frac{|\vec{A} \times \vec{B}|}{|\vec{AB}|} \]

### Step 7: Substitute for \( \vec{AB} \)
Substituting \( \vec{AB} = \vec{B} - \vec{A} \):
\[ |CM| = \frac{|\vec{A} \times \vec{B}|}{|\vec{B} - \vec{A}|} \]

### Final Result
Thus, the perpendicular distance of point \( \vec{C} \) from the line joining \( \vec{A} \) and \( \vec{B} \) is:
\[ |CM| = \frac{|\vec{A} \times \vec{B}|}{|\vec{B} - \vec{A}|} \]

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