Find the value of `lamda` so that the straight lines: `(1-x)/(3)= (7y -14)/(2 lamda)= (z-3)/(2) and (7-7x)/(3 lamda)= (y-5)/(1)= (6-z)/(5)` are at right angles.

Answer

To find the value of \( \lambda \) such that the given straight lines are at right angles, we will follow these steps:

### Step 1: Write the equations of the lines in standard form
The given equations are:
\[ \frac{1-x}{3} = \frac{7y - 14}{2\lambda} = \frac{z - 3}{2} \]
and
\[ \frac{7 - 7x}{3\lambda} = \frac{y - 5}{1} = \frac{6 - z}{5} \]

We can rewrite these equations in the parametric form.

For the first line, let \( t \) be the parameter:
\[ x = 1 - 3t, \quad y = 2 + \lambda t, \quad z = 3 + 2t \]

For the second line, let \( s \) be the parameter:
\[ x = 1 - \frac{3\lambda s}{7}, \quad y = 5 + s, \quad z = 6 - 5s \]

### Step 2: Find the direction ratios of the lines
The direction ratios of the first line are:
\[ (-3, 2\lambda, 2) \]
The direction ratios of the second line are:
\[ \left(-\frac{3\lambda}{7}, 1, -5\right) \]

### Step 3: Set up the dot product condition for perpendicularity
For the lines to be at right angles, the dot product of their direction ratios must equal zero:
\[ (-3) \left(-\frac{3\lambda}{7}\right) + (2\lambda)(1) + (2)(-5) = 0 \]

### Step 4: Simplify the equation
Calculating the dot product:
\[ \frac{9\lambda}{7} + 2\lambda - 10 = 0 \]
Combine the terms:
\[ \frac{9\lambda + 14\lambda}{7} - 10 = 0 \]
This simplifies to:
\[ \frac{23\lambda}{7} - 10 = 0 \]

### Step 5: Solve for \( \lambda \)
Rearranging gives:
\[ \frac{23\lambda}{7} = 10 \]
Multiplying both sides by 7:
\[ 23\lambda = 70 \]
Dividing by 23:
\[ \lambda = \frac{70}{23} \]

### Final Answer
The value of \( \lambda \) for which the two lines are at right angles is:
\[ \lambda = \frac{70}{23} \]