Find the foot of perpendicular and image of point (1,2,1) along the line
x−3/1 = y+1/2 = z−1/3 = λ.

To find the foot of the perpendicular and the image of the point (1, 2, 1) along the given line, we need to follow these steps:

The equation of the line is given in symmetric form: x − 3 / 1 = y + 1 / 2 = z − 1 / 3 = λ.

This line can be expressed in parametric form as: x = 1λ + 3, y = 2λ − 1, z = 3λ + 1.

Let's denote the point P(1, 2, 1) as P(x₁, y₁, z₁), which has the coordinates (1, 2, 1).

Step 1: Equation of the line joining the point P to the line
The direction ratios of the line are 1, 2, and 3. We will write the equation of the line joining P to a point on the given line (denoted by Q) in terms of the parameter λ.

The coordinates of point Q are (x, y, z) = (1λ + 3, 2λ − 1, 3λ + 1).

The vector from P(1, 2, 1) to Q(1λ + 3, 2λ − 1, 3λ + 1) is given by: PQ = [(1λ + 3 − 1), (2λ − 1 − 2), (3λ + 1 − 1)] = (λ + 2, 2λ − 3, 3λ).

Step 2: Perpendicular condition
For the line PQ to be perpendicular to the given line, the dot product of the direction vector of the given line (which is (1, 2, 3)) and the vector PQ must be zero.

Dot product of (1, 2, 3) and (λ + 2, 2λ − 3, 3λ): (1)(λ + 2) + (2)(2λ − 3) + (3)(3λ) = 0.

Expanding: λ + 2 + 4λ − 6 + 9λ = 0, 14λ − 4 = 0, 14λ = 4, λ = 4 / 14 = 2 / 7.

Step 3: Coordinates of the foot of the perpendicular
Now, substitute λ = 2 / 7 into the parametric equations of the line to find the coordinates of the foot of the perpendicular:

x = 1(2/7) + 3 = 3 + 2/7 = 23/7, y = 2(2/7) − 1 = 4/7 − 1 = −3/7, z = 3(2/7) + 1 = 6/7 + 1 = 13/7.

So, the foot of the perpendicular is at the point (23/7, −3/7, 13/7).

Step 4: Image of the point
To find the image of the point P with respect to the line, we use the formula: Image of P = 2 × Foot of perpendicular − P.

Substitute the coordinates: Image = 2 × (23/7, −3/7, 13/7) − (1, 2, 1).

Multiply by 2: (46/7, −6/7, 26/7) − (1, 2, 1) = (46/7 − 7/7, −6/7 − 14/7, 26/7 − 7/7), Image = (39/7, −20/7, 19/7).

Final Answer:
The foot of the perpendicular is at the point (23/7, −3/7, 13/7), and the image of the point (1, 2, 1) along the line is at (39/7, −20/7, 19/7).