Distance of the point `P(vecp)` from the line `vecr=veca+lamdavecb` is
(a)`|(veca-vecp)+(((vecp-veca).vecb)vecb)/(|vecb|^(2))|` (b)`|(vecb-vecp)+(((vecp-veca).vecb)vecb)/(|vecb|^(2))|`
(c)`|(veca-vecp)+(((vecp-vecb).vecb)vecb)/(|vecb|^(2))|` (d)none of these

Options

A. `|(veca=vecp)+(((vecp-veca).vecb)vecb)/(|vecb|^(2))|`
B. `|(vecb-vecp)+(((vecp-veca).vecb)vecb)/(|vecb|^(2))|`
C. `|(veca-vecp)+(((vecp-veca).vecb)vecb)/(|vecb|^(2))|`
D. none of these

Explanation

To find the distance of the point \( P \) represented by the vector \( \vec{p} \) from the line given by \( \vec{r} = \vec{a} + \lambda \vec{b} \), we can follow these steps:

### Step 1: Understand the line and point
The line can be represented as \( \vec{r} = \vec{a} + \lambda \vec{b} \), where \( \vec{a} \) is a point on the line, \( \lambda \) is a scalar parameter, and \( \vec{b} \) is the direction vector of the line. The point \( P \) is represented by the vector \( \vec{p} \).

**Hint:** Identify the components of the line and the point clearly.

### Step 2: Find a point \( D \) on the line
For some value of \( \lambda_1 \), the point \( D \) on the line can be expressed as:
\[ \vec{D} = \vec{a} + \lambda_1 \vec{b} \]

**Hint:** Remember that \( D \) is a point on the line defined by \( \vec{a} \) and \( \vec{b} \).

### Step 3: Establish the perpendicularity condition
The line segment \( AP \) (from point \( A \) to point \( P \)) is perpendicular to the line. Therefore, the vector \( \vec{D} - \vec{p} \) must be perpendicular to \( \vec{b} \). This gives us the condition:
\[ (\vec{D} - \vec{p}) \cdot \vec{b} = 0 \]

**Hint:** Use the dot product to express the condition of perpendicularity.

### Step 4: Substitute \( \vec{D} \) into the perpendicularity condition
Substituting \( \vec{D} \) into the equation gives:
\[ (\vec{a} + \lambda_1 \vec{b} - \vec{p}) \cdot \vec{b} = 0 \]
This simplifies to:
\[ (\vec{a} - \vec{p}) \cdot \vec{b} + \lambda_1 |\vec{b}|^2 = 0 \]

**Hint:** Rearranging this equation will help you find \( \lambda_1 \).

### Step 5: Solve for \( \lambda_1 \)
From the equation, we can solve for \( \lambda_1 \):
\[ \lambda_1 = \frac{(\vec{p} - \vec{a}) \cdot \vec{b}}{|\vec{b}|^2} \]

**Hint:** Make sure to pay attention to the signs when rearranging.

### Step 6: Find the coordinates of point \( D \)
Now substitute \( \lambda_1 \) back into the equation for \( \vec{D} \):
\[ \vec{D} = \vec{a} + \frac{(\vec{p} - \vec{a}) \cdot \vec{b}}{|\vec{b}|^2} \vec{b} \]

**Hint:** This gives you the coordinates of point \( D \) in terms of \( \vec{a} \), \( \vec{b} \), and \( \vec{p} \).

### Step 7: Calculate the distance \( AP \)
The distance \( AP \) can be expressed as:
\[ AP = |\vec{D} - \vec{p}| \]
Substituting \( \vec{D} \) gives:
\[ AP = |\left(\vec{a} - \vec{p}\right) + \frac{(\vec{p} - \vec{a}) \cdot \vec{b}}{|\vec{b}|^2} \vec{b}| \]

**Hint:** This expression represents the distance from point \( P \) to the line.

### Step 8: Final expression
Thus, the distance of point \( P \) from the line is:
\[ |\left(\vec{a} - \vec{p}\right) + \frac{(\vec{p} - \vec{a}) \cdot \vec{b}}{|\vec{b}|^2} \vec{b}| \]

This matches option (a):
\[ |\left(\vec{a} - \vec{p}\right) + \frac{(\vec{p} - \vec{a}) \cdot \vec{b}}{|\vec{b}|^2} \vec{b}| \]

### Conclusion
The correct answer is (a).

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