Distance of the point `P(vecc)` from the line `vecr=veca+lamdavecb` is

Options

A. `(|(vecc-veca)xxvecb|)/(|vecb|)`
B. `(|(vecc-veca).vecb|)/(|vecb|)`
C. `(|(vecc-veca)xxvecb|)/(|vecb|^(2))`
D. none of these

Explanation

To find the distance of the point \( P(\vec{c}) \) from the line defined by \( \vec{r} = \vec{a} + \lambda \vec{b} \), we will follow these steps:

### Step 1: Identify the components
Let:
- \( \vec{P} = \vec{c} \) (the point from which we want to find the distance)
- \( \vec{A} = \vec{a} \) (a point on the line)
- \( \vec{B} = \vec{b} \) (the direction vector of the line)

### Step 2: Find the vector from point A to point P
Calculate the vector \( \vec{AP} \):
\[ \vec{AP} = \vec{c} - \vec{a} \]

### Step 3: Project vector AP onto vector B
To find the projection of \( \vec{AP} \) onto \( \vec{B} \), we use the formula:
\[ \text{Projection of } \vec{AP} \text{ onto } \vec{B} = \frac{\vec{AP} \cdot \hat{b}}{|\vec{b}|^2} \vec{b} \]
where \( \hat{b} = \frac{\vec{b}}{|\vec{b}|} \) is the unit vector in the direction of \( \vec{b} \).

### Step 4: Calculate the length of the projection
The length of the projection \( QM \) (where \( Q \) is the foot of the perpendicular from point \( P \) to the line) is given by:
\[ QM = \frac{\vec{AP} \cdot \hat{b}}{|\vec{b}|} \]

### Step 5: Find the perpendicular distance
The distance \( d \) from point \( P \) to the line is given by:
\[ d = |\vec{AP}|^2 - |QM|^2 \]
Using the Pythagorean theorem, we can express this as:
\[ d = \sqrt{|\vec{AP}|^2 - |QM|^2} \]

### Step 6: Substitute and simplify
Substituting the expressions for \( |\vec{AP}| \) and \( |QM| \):
\[ d = \sqrt{|\vec{c} - \vec{a}|^2 - \left(\frac{(\vec{c} - \vec{a}) \cdot \vec{b}}{|\vec{b}|}\right)^2} \]

### Step 7: Final expression
The final expression for the distance from point \( P \) to the line is:
\[ d = \frac{|\vec{c} - \vec{a} \times \vec{b}|}{|\vec{b}|} \]

### Summary
Thus, the distance of the point \( P(\vec{c}) \) from the line \( \vec{r} = \vec{a} + \lambda \vec{b} \) is given by:
\[ d = \frac{|\vec{c} - \vec{a} \times \vec{b}|}{|\vec{b}|} \]